PhD thesis - IAS

N o d’ordre: 9910
THÈSE
Présentée pour obtenir
LE GRADE DE DOCTEUR EN SCIENCES
DE L’UNIVERSITÉ PARIS-SUD XI
Spécialité: Mathématiques
par
Abed Bounemoura
Stabilité et instabilité des systèmes
hamiltoniens presque-intégrables.
Soutenue le 22 Septembre 2010 devant la Commission d’examen:
M. Patrick BERNARD
M. Håkan ELIASSON
M. Pierre LOCHAK
M. Jean-Pierre MARCO (Directeur de thèse)
M. Laurent NIEDERMAN (Directeur de thèse)
M. Jean-Christophe YOCCOZ (Président du jury)
Rapporteurs:
M. Patrick BERNARD
M. Vadim KALOSHIN

Thèse préparée au
Département de Mathématiques d’Orsay
Laboratoire de Mathématiques (UMR 8628), Bât. 425
Université Paris-Sud 11
91 405 Orsay CEDEX

Résumé
Dans cette thèse, on s’intéresse à diverses questions concernant la stabilité et l’instabilité
des systèmes hamiltoniens presque-intégrables. Elle se divise en quatre parties et huit chapitres.
Dans une première partie, on donne une introduction informelle aux systèmes hamiltoniens
et à la théorie des perturbations des systèmes hamiltoniens intégrables dans le premier chapitre,
puis dans le second chapitre, on expose les résultats présentés dans cette thèse.
Une seconde partie est consacrée à des résultats de stabilité. Dans le troisième chapitre,
on donne une nouvelle preuve du théorème de stabilité exponentielle de Nekhoroshev dans le
cas générique pour un système analytique. Elle n’utilise que des compositions de moyennisations
périodiques, et elle évite donc le fameux problème des petits diviseurs. Dans le quatrième
chapitre, on utilise cette approche pour en déduire des nouveaux résultats de stabilité exponentielle
et super-exponentielle au voisinage des points fixes elliptiques, des tores lagrangiens
invariants quasi-périodiques et plus généralement des tores invariants quasi-périodiques linéairement
stables, isotropes et réductibles. Enfin, dans le cinquième chapitre, on établit un résultat
de stabilité polynomiale si le système est seulement de différentiabilité finie, dans le cas où la
partie intégrable est quasi-convexe.
Dans une troisième partie, on étudie la frontière entre la stabilité et l’instabilité. Dans le
sixième chapitre, pour un système quasi-convexe analytique ou de classe Gevrey, on améliore
l’exposant de stabilité en étudiant la géométrie des résonances simples. On obtient ainsi un
temps de stabilité encore plus proche des temps d’instabilité connus, et qui doit certainement
être optimal.
Enfin, dans une quatrième partie, on s’intéresse à la construction de certains exemples
d’instabilité. Dans un septième chapitre, on construit un nouvel exemple d’un système a priori
instable qui possède une solution qui dérive avec un temps optimal. Notre approche est basée sur
la dynamique symbolique engendrée par l’intersection transverse des variétés stable et instable
d’une variété normalement hyperbolique. Dans le huitième et dernier chapitre, on construit
également un exemple de système presque-intégrable, dont la taille de la perturbation ne tend
vers zéro que lorsque le nombre de degrés de libertés tend vers l’infini, avec une solution qui
dérive en temps polynomial. Cela donne en particulier de nouvelles contraintes sur le seuil de
validité des résultats de stabilité exponentielle.
La première partie est écrite en français, les autres en anglais.
Mots-clefs : Systèmes dynamiques, Systèmes hamiltoniens, Théorie des perturbations, Théorie
de Nekhoroshev, Diffusion d’Arnold.

Stability and instability of near-integrable Hamiltonian systems.
Abstract
In this thesis, we study various questions concerning the stability and instability of nearintegrable
Hamiltonian systems. It is divided into four parts and eight chapters.
In a first part, we give an informal introduction to Hamiltonian systems and to the perturbation
theory of integrable Hamiltonian systems in the first chapter, and then, in the second
chapter, we state our results.
A second part is devoted to stability results. In the third chapter, we give a new proof of the
exponential stability theorem of Nekhoroshev in the generic case for an analytic system. Our
method uses only composition of periodic averaging, and therefore it avoids the small divisors
problem. Then, in the fourth chapter, we take advantage of this approach to obtain new
results of exponential and super-exponential stability in the neighbourhood of elliptic fixed
points, invariant Lagrangian quasi-periodic tori and more generally invariant linearly stable
quasi-periodic tori, which are isotropic and reducible. In the fifth chapter, for a quasi-convex
integrable Hamiltonian system, we also prove a result of polynomial stability in the case where
the system is only finitely differentiable.
A third part lies between stability and instability. In the sixth chapter, for a quasi-convex
system which is analytic or Gevrey, we improve the stability exponent by studying the geometry
of simple resonances. Thus we obtain a time of stability which is closer to the known instability
times, and which is certainly optimal.
In the fourth part, we will construct examples of unstable Hamiltonian systems. First,
in the seventh chapter, we give a new example of an a priori unstable system which has a
drifting orbit with an optimal time of instability. Our method uses the symbolic dynamics
created by the transverse intersection between the stable and unstable manifolds of a normally
hyperbolic invariant manifold. In the eighth and last chapter, we also construct an example
of a near-integrable Hamiltonian system, for which the size of the perturbation goes to zero
only when the number of degrees of freedom goes to infinity, and which has an orbit drifting
in a polynomial time. In particular, this gives a new constraint on the threshold of validity for
exponential stability results.
The first part is written in French, and the others in English.
Keywords : Dynamical Systems, Hamiltonian Systems, Perturbation Theory, Nekhoroshev
Theory, Arnold Diffusion.

Béton style ...
Quelque soit l’âge, le sexe, le niveau d’étude
La vie fait de nous ce qu’on est, il y’a pas de quoi garder rancune,
Être déboussolé ou plein d’amertume,
Tant qu’on a la volonté, la liberté de choisir, pas d’inquiétude,
Nous apprenons jour et nuit la vie entre ciel et terre,
On peut répondre non et oui,
Aimer mère et père,
Évaluer le pour et le contre,
Se retrouver haut et bas,
Perdre ou retrouver son compte lorsqu’on est entre le bien et le mal,
Ce que la vie offre à la vie
Ne nourrit pas tout le temps l’espoir,
On croit aux amours, tolérances, douceurs et joies,
Y’a aussi le côté morose,
Haine, humeur noire, angoisse, fureur, crainte, tristesse, et désespoir,
Chacun choisit ses valeurs, sa manière d’être et de voir,
Son code moral, son mode de vie, son clan, son devoir,
J’dis ces trucs, histoire de faire savoir
Ce que c’est de ne savoir que faire à part se servir de son savoir.
Le Rat Luciano, Fonky Family. Entre deux feux, Art de rue, 2001.

À ma mère, à toute ma famille

Remerciements
À mes deux excellents directeurs de thèse, Laurent Niederman et Jean-Pierre Marco,
sans qui cette thèse n’existerait pas. Cette page serait beaucoup trop courte pour tout
les remerciements professionnels et personnels que je leur dois. Ce fût un immense plaisir
de travailler avec eux ces trois dernières années, et j’espère bien que cela continuera.
À Patrick Bernard et Vadim Kaloshin, pour l’intérêt qu’ils ont portés à ce travail
en leur qualité de rapporteur. À Pierre Lochak, pour être à la source de la plupart des
mathématiques de cette thèse. À Håkan Eliasson et Jean-Christophe Yoccoz, pour le
très grand honneur qu’ils me font en étant présents dans ce jury.
À Frédéric Le Roux, pour m’avoir initié avec beaucoup de gentillesse et d’efficacité
au monde de la recherche mathématiques; je lui dois entièrement le livre [Bou08] issu
d’un stage de Master effectué sous sa direction, et que j’ai rédigé durant ma première
année de thèse. Merci aussi à Étienne Ghys d’avoir proposé que ce travail soit publié.
Au Laboratoire Mathématiques d’Orsay, à Frédéric Le Roux, François Béguin, Sylvain
Crovisier, Pierre Pansu, David Harari et Frédéric Paulin, mais aussi aux meilleurs
secrétaires du monde, Valérie Blandin-Lavigne et Martine Justin.
À l’Institut Mathématiques de Jussieu, à Jacques Féjoz, Laurent Lazzarini, Maylis
Irigoyen et à tous les participants du séminaire "Géométrie Hamiltonienne", à Marcelline
Prosper-Cojande pour sa compétence et sa gentillesse.
À l’Observatoire de Paris, à tous les membres de l’équipe ASD de l’IMCCE, avec des
remerciements très spéciaux aux "deux Alain", Alain Albouy et Alain Chenciner, mais
aussi à Phillipe Robutel, Jacques Laskar et David Sauzin.
À Laurent Niederman, Jean-Pierre Marco, Alain Chenciner, Alain Albouy, Frédéric
Le Roux, Tere Seara, Ernest Fontich, Enrique Pujals mais aussi Pierre Berger et Alfonso
Sorrentino, pour divers soutiens. À El Houcein El Abdalaoui, Jean-François Quint, Sylvain
Crovisier et Andrea Venturelli pour leur invitation à des séminaires "extérieurs" et
leur excellent accueil. À Bassam Fayad, son cours de Master sur la théorie KAM m’a
orienté vers les systèmes hamiltoniens. À Claudio Murolo, pour avoir été le premier à
croire en mes capacités mathématiques et pour m’avoir encouragé dans cette voie.
À tous mes amis doctorants et jeunes docteurs, qu’ils viennent d’Orsay, de
Chevaleret-Jussieu, de l’Observatoire de Paris ou des quatre coins du monde. C’est
le moment difficile et injuste où je ne citerai pas de noms alors que je leur dois beaucoup;
j’ai eu la chance d’être entouré de brillants (apprentis) mathématiciens à qui je
pouvais poser mes questions, ainsi que d’un très grand expert en Latex. J’adresserais
mes remerciements à chacun de vive voix.
Aux Mathématiques pour ce qu’elles m’ont apporté, à savoir une vie agréable ces
trois dernières années avec la possibilité d’assouvir ma passion pour les voyages. En
retour, j’espère pouvoir un jour apporter quelque chose aux Mathématiques.
À tous mes amis parisiens et amies parisiennes.
À tous mes potes du quartier, je n’oublie pas d’où je viens.
À celle dont je ne connais pas le nom, avec un brin d’optimisme elle le saura un jour.

Table des matières
I Introduction, résultats et questions 15
1 Introduction 17
1.1 Systèmes hamiltoniens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Systèmes hamiltoniens intégrables . . . . . . . . . . . . . . . . . . . . . . 22
1.3 Théorie classique des perturbations . . . . . . . . . . . . . . . . . . . . . 27
1.4 Théorèmes de stabilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5 Exemples d’instabilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.6 Au voisinage d’un tore invariant linéairement stable . . . . . . . . . . . . 46
2 Résultats et questions 51
2.1 Résultats de stabilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 De la stabilité à l’instabilité . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Résultats d’instabilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
II Results of stability 57
3 Generic exponential stability without small divisors 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.A Proof of the normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.B SDM functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4 Generic super-exponential stability for invariant tori 99
4.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Proof of Theorem 4.1 and Theorem 4.2 . . . . . . . . . . . . . . . . . . . 104
4.3 Further results and comments . . . . . . . . . . . . . . . . . . . . . . . . 108
4.A Generic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Polynomial stability for C k quasi-convex Hamiltonian systems 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Analytical part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
III From stability to instability 129
6 Improved exponential stability for quasi-convex Hamiltonian systems131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 The analytic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 The Gevrey case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
IV Results of instability 149
7 Optimal time of instability for a priori unstable Hamiltonian systems151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3 Construction of the perturbation . . . . . . . . . . . . . . . . . . . . . . 155
7.4 Construction of a symbolic dynamic . . . . . . . . . . . . . . . . . . . . . 159
7.5 Construction of a pseudo-orbit . . . . . . . . . . . . . . . . . . . . . . . . 176
7.6 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.A Time-energy coordinates for the pendulum . . . . . . . . . . . . . . . . . 191
8 Time of instability for high-dimensional Hamiltonian systems 193
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.3 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.A Gevrey functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Références 209

Première partie
Introduction, résultats et questions
Sommaire
1 Introduction 17
1.1 Systèmes hamiltoniens . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.1 Équations de la mécanique classique . . . . . . . . . . . . . . 17
1.1.2 Systèmes hamiltoniens sur une variété . . . . . . . . . . . . . 19
1.1.3 Quelques propriétés générales . . . . . . . . . . . . . . . . . . 21
1.2 Systèmes hamiltoniens intégrables . . . . . . . . . . . . . . . . . . . . 22
1.2.1 Structure des systèmes intégrables . . . . . . . . . . . . . . . 23
1.2.2 Dynamique des systèmes intégrables . . . . . . . . . . . . . . 26
1.3 Théorie classique des perturbations . . . . . . . . . . . . . . . . . . . 27
1.3.1 Principe de moyennisation . . . . . . . . . . . . . . . . . . . . 28
1.3.2 Théorie des formes normales . . . . . . . . . . . . . . . . . . . 29
1.4 Théorèmes de stabilité . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.1 Théorie KAM . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.2 Théorie de Nekhoroshev . . . . . . . . . . . . . . . . . . . . . 36
1.5 Exemples d’instabilité . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5.1 Le mécanisme d’Arnold . . . . . . . . . . . . . . . . . . . . . 40
1.5.2 Résonances simples et résonances multiples . . . . . . . . . . 43
1.5.3 Temps d’instabilité . . . . . . . . . . . . . . . . . . . . . . . . 45
1.6 Au voisinage d’un tore invariant linéairement stable . . . . . . . . . . 46
1.6.1 Au voisinage d’un tore lagrangien quasi-périodique . . . . . . 46
1.6.2 Au voisinage d’un point fixe elliptique . . . . . . . . . . . . . 48
2 Résultats et questions 51
2.1 Résultats de stabilité . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.1 Cas générique . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.2 Cas quasi-convexe . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 De la stabilité à l’instabilité . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.1 Cas analytique . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.2 Cas Gevrey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Résultats d’instabilité . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.1 Cas a priori instable . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.2 Cas non perturbatif . . . . . . . . . . . . . . . . . . . . . . . 56
15

1.1 - Systèmes hamiltoniens 17
1 Introduction
Dans cette première partie on donne une introduction informelle aux systèmes hamiltoniens
et à la théorie des perturbations des systèmes hamiltoniens intégrables. Le
lecteur averti pourra passer directement au chapitre suivant présentant les résultats
de cette thèse. De plus, il retrouvera par la suite quelques rappels plus condensés et
spécialisés dans l’introduction de chaque chapitre. Une référence générale pour cette
introduction est [AKN06].
1.1 Systèmes hamiltoniens
On commence par quelques rappels généraux sur les systèmes hamiltoniens.
1.1.1 Équations de la mécanique classique
1.1.1.1. Le point de départ de la théorie des systèmes hamiltoniens, et plus généralement
de toute la théorie des systèmes dynamiques, est l’étude de certaines équations
différentielles d’ordre deux de la forme
¨q = −∇V (q) (1)
où q ∈ R n , V : R n → R est une fonction lisse (de classe au moins C 2 ), et où le gradient
∇ dépend d’une structure euclidienne sur R n . Ces équations, décrivant l’évolution d’un
point matériel (de position q et de masse unité) dans un champ de force conservatif
(dérivant du potentiel V ), sont bien connues pour modéliser de nombreux problèmes issus
de la mécanique classique et céleste. Bien entendu, sauf cas exceptionnels, il n’est pas
possible "d’intégrer" ces équations, et suivant Poincaré, il est devenu courant d’étudier
le comportement asymptotique des solutions.
On peut facilement transformer cette équation d’ordre deux sur Rn (l’espace des
configurations) en un système d’équations d’ordre un sur R2n (l’espace des phases),
ceci en introduisant la vitesse (ou l’impulsion) p = ˙q ∈ Rn comme nouvelle variable.
L’équation (1) est alors équivalente au système
˙q = p
(2)
˙p = −∇V (q).
Un calcul direct montre que l’énergie totale du système, c’est-à-dire la fonction
H(q, p) = 1
2 p2 + V (q),
est constante durant l’évolution, et que le système s’écrit encore
˙q = ∂pH(q, p)
˙p = −∂qH(q, p).
Il s’agit de la formulation hamiltonienne de l’équation (1), et l’énergie totale H est
également appelée hamiltonien.
(3)

18 Introduction
En réalité, ces équations sont de nature plus intrinsèque. En effet, si J0 désigne la
structure complexe standard de R2n , c’est-à-dire
0 In
J0 =
∈ M2n(R)
−In 0
dans la base canonique (In étant la matrice identité de taille n), et si Ω0 est la forme
bilinéaire alternée définie par la matrice J0, alors les solutions des équations (3) ne sont
rien d’autre que les courbes intégrales du champ de vecteurs XH défini par
d(q,p)H.Y = Ω0(XH(q, p), Y ),
pour tout point (q, p) ∈ R 2n et tout vecteur Y ∈ R 2n . Si on utilise le produit intérieur,
cela donne iXH Ω0 = dH. Notons que cette définition a bien un sens puisque Ω0 est non
dégénérée et induit donc un isomorphisme entre R 2n (ou plus exactement son espace
tangent) et son dual (son espace cotangent). Ainsi, les équations (3) s’écrivent encore
( ˙q, ˙p) = XH(q, p). (4)
On dit que XH est le champ de vecteurs hamiltonien, ou gradient symplectique, associé
à H. Si ∇ désigne le gradient relatif à la structure euclidienne canonique de R 2n , on a
l’égalité
XH = J0∇H
que l’on peut interpréter géométriquement de la manière suivante : un champ hamiltonien
est un champ de gradient usuel que l’on fait "tourner d’un quart de tour".
1.1.1.2. Donnons maintenant l’exemple du fameux problème des N corps, qui constitue
la motivation originelle. On considère le mouvement de N particules, de position
q1, . . .,qN et de masses m1, . . .,mN, évoluant dans un espace euclidien E et soumises à
l’interaction gravitationnelle : l’attraction entre deux particules est proportionnelle au
produit des masses et à l’inverse du carré de la distance mutuelle. On pourra bien sûr
penser au système solaire où, en admettant que les planètes sont à symétrie sphérique,
on peut, selon Newton, les remplacer par leurs centres de gravité en y concentrant leurs
masses. En notant G la constante gravitationnelle, les équations s’écrivent
mi¨qi = −G
i=j
mimj
qi − qj
||qi − qj|| 3,
et elles sont bien définies sur l’ouvert "sans collisions"
En introduisant le potentiel
les équations s’écrivent sous la forme
{q = (q1, . . .,qN) ∈ E N | qi = qj, i = j}.
U(q) = G
i=j
¨q = −∇U(q)
mimj
||qi − qj|| ,

1.1 - Systèmes hamiltoniens 19
où le gradient est pris relativement au produit scalaire euclidien sur E N pondéré par les
masses, c’est-à-dire q.q ′ = N
i=1 miqiq ′ i. Finalement le hamiltonien est donné par
H = 1
2
N
mi|| ˙qi|| 2 − G mimj
||qi − qj|| .
i=1
Il est bien connu, et on le verra par la suite, qu’il est possible de résoudre ces équations
pour N = 2, mais que c’est loin d’être le cas pour N ≥ 3.
Il convient maintenant d’étendre l’étude dans deux directions. D’une part, dans les
exemples concrets, comme le précédent, l’espace des configurations n’est jamais R n tout
entier mais plutôt un ouvert ou une sous-variété; il s’agit donc de se placer sur une
variété abstraite. D’autre part, d’un point de vue théorique évident il est préférable de
considérer une fonction hamiltonienne arbitraire.
1.1.2 Systèmes hamiltoniens sur une variété
1.1.2.1. Les systèmes hamiltoniens sur les variétés peuvent se définir dans deux cadres
distincts : celui des variétés symplectiques et celui plus général des variétés de Poisson.
On va essentiellement s’intéresser au premier, le plus classique, et mentionner brièvement
le second.
Définition 1.1. Soit M une variété lisse. Une structure symplectique sur M est une
2-forme différentielle Ω qui est :
- non dégénérée, c’est-à-dire que Ωx est non dégénérée sur TxM pour tout x ∈ M ;
- fermée, c’est-à-dire que dΩ = 0.
On dit alors que le couple (M, Ω) est une variété symplectique.
Puisque Ω est non dégénérée, on voit facilement que M est nécessairement de dimension
paire, et qu’elle est orientable (Ω n = Ω ∧ · · · ∧ Ω est partout non nulle). Sous
cette même hypothèse, on peut donner la définition suivante.
Définition 1.2. Soit (M, Ω) une variété symplectique et H : M → R une fonction lisse.
Le champ de vecteurs hamiltonien XH est défini par iXHΩ = dH, c’est-à-dire
pour tout vecteur tangent Y ∈ TxM.
i=j
dxH.Y = Ωx(XH(x), Y ), x ∈ M,
Notons que la définition d’un champ de vecteurs hamiltonien n’utilise que l’hypothèse
de non-dégénérescence de Ω; cependant, on verra dans la suite en quoi l’hypothèse de
fermeture sur Ω est importante.
Sur toute variété symplectique (M, Ω) on peut trouver une structure presque complexe
J (c’est-à-dire un endomorphisme du fibré tangent dont le carré vaut moins l’identité)
et une structure riemannienne g compatible avec Ω dans le sens où g(X, Y ) =
Ω(X, JY ). Alors, on a encore la formule XH = J∇gH où ∇g est le gradient relatif à g.
1.1.2.2. Donnons maintenant deux exemples de variétés symplectiques. Le premier, et
le plus simple, est celui d’un espace vectoriel V muni d’une forme bilinéaire alternée non

20 Introduction
dégénérée. Il n’est pas difficile de construire alors une base de V (dite symplectique)
dans laquelle la forme bilinéaire se représente par la matrice J0 précédemment définie.
En d’autres termes, tout espace vectoriel symplectique est linéairement isomorphe
à R 2n muni de sa structure symplectique standard Ω0. Pour une variété symplectique
quelconque, le résultat précédent reste vrai localement par le théorème classique de Darboux
: on peut toujours trouver localement des coordonnées (q, p), dites symplectiques,
dans lesquelles la forme Ω s’exprime par
Ω = dq ∧ dp.
C’est ici qu’intervient l’hypothèse de fermeture sur Ω (cette dernière est donc une hypothèse
de "courbure nulle"), et sans elle il n’est donc pas possible de faire des calculs
explicites sur une variété symplectique abstraite.
Le second, et le plus important en ce qui nous concerne, est celui du fibré cotangent
d’une variété N : c’est l’espace des phases des équations de la mécanique classique.
On dispose sur M = T ∗N d’une 1-forme différentielle canonique σ que l’on peut définir
de la manière suivante : si q = (q1, . . .,qn) sont des coordonnées locales sur N et
p = (p1, . . .,pn) les coordonnées locales associées sur la fibre T ∗ q N, alors σ = pdq. On
peut vérifier que cela ne dépend pas du choix des coordonnées (ou bien on peut définir
σ de manière intrinsèque, quoique cela ne soit pas très instructif), et que σ vérifie
l’identité α ∗ σ = α pour toute 1-forme α sur M. La structure symplectique canonique de
T ∗ N est alors définie par Ω0 = −dσ, c’est-à-dire Ω0 = dq ∧ dp en coordonnées. Notons
que les fibrés cotangents ne sont jamais compacts, et qu’ils ont la propriété supplémentaire
immédiate d’être des variétés symplectiques exactes, c’est-à-dire que leur forme
symplectique possède une primitive.
1.1.2.3. Pour conclure, mentionnons le cadre des variétés de Poisson, qui généralise celui
des variétés symplectiques dans le sens où l’on autorise la forme fermée Ω à "dégénérer".
Commençons par une définition plus simple.
Définition 1.3. Une structure de Poisson sur M est la donnée d’une application
{., .} : C ∞ (M) × C ∞ (M) −→ C ∞ (M)
bilinéaire, symétrique et qui vérifie pour tout f, g, h ∈ C ∞ (M), les relations
- {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0;
- {fg, h} = f{g, h} + {f, h}g.
On appelle crochet de Poisson une telle application, et les deux relations sont souvent
appelées identité de Jacobi et identité de Leibniz. On dit alors que (C ∞ (M), {., .}) est
une algèbre de Poisson, c’est-à-dire une algèbre de Lie telle que pour toute fonction H
sur M, l’application {., H} est une dérivation de C ∞ (M). En particulier, si M est de
dimension finie, il existe un unique champ de vecteurs XH associé à cette dérivation, et
c’est par définition le champ de vecteurs hamiltonien engendré par H.
Une variété symplectique (M, Ω) est naturellement une variété de Poisson si l’on
pose
{f, g} = Xgf = Ω(Xf, Xg),
et l’identité de Jacobi est alors équivalente à la fermeture de Ω ou bien encore à l’existence
locale de coordonnées de "Poisson" (q, p) dans lesquelles le crochet s’exprime de

1.1 - Systèmes hamiltoniens 21
manière classique
{f, g} =
n
i=1
∂qif∂pi g − ∂pif∂qi g.
Les variétés de Poisson sont donc plus générales, mais pour mieux saisir la différence il
est nécessaire de les définir en termes un peu plus sophistiqués de géométrie différentielle.
En effet, la définition du crochet de Poisson implique que l’on peut l’écrire
{f, g} = ˜ Ω(df, dg),
pour un certain champ de bi-vecteurs ˜ Ω ∈ Λ 2 TM vérifiant [ ˜ Ω, ˜ Ω] = 0 (ou [., .] est le
crochet de Schouten, qui généralise le crochet de Lie des champs de vecteurs). Le cas où
˜Ω est en tout point non dégénérée nous donne par dualité une structure symplectique
Ω. Dans le cas général, l’ensemble des points où le rang est constant (nécessairement
pair) forme une sous-variété qui hérite naturellement d’une structure symplectique, et
l’ensemble de ces sous-variétés forme un "feuilletage" symplectique (où la dimension des
feuilles varient).
1.1.3 Quelques propriétés générales
1.1.3.1. On a déjà vu que la définition d’un champ de vecteurs hamiltonien ne requiert
que la donnée d’une fonction H, le hamiltonien (la structure symplectique sous-jacente
étant fixée), ce qui les distingue beaucoup des champs de vecteurs arbitraires. Si l’on
pense en termes d’équations différentielles, il est souvent pratique d’effectuer des changements
de variables pour simplifier l’étude, dans le cas des systèmes hamiltoniens cette
opération est très facile. Commençons par préciser la notion de changement de variables
dans ce contexte, où il faut bien sûr préserver la forme hamiltonienne des équations.
Définition 1.4. On dit qu’une application Φ : (M, Ω) → (M, Ω) est symplectique si
Φ ∗ Ω = Ω.
De manière équivalente, Φ préserve le crochet de Poisson dans le sens où
{f ◦ Φ, g ◦ Φ} = {f, g},
pour toutes fonctions f, g ∈ C ∞ (M). Lorsque Φ est un difféomorphisme symplectique
et XH un champ de vecteurs hamiltonien, effectuer un changement de variables à l’aide
de Φ revient à étudier le champ de vecteurs Φ ∗ XH.
Proposition 1.5 (Changement de variables). Si Φ est un difféomorphisme symplectique,
alors Φ ∗ XH = XH◦Φ.
Autrement dit, pour transformer les champs de vecteurs hamiltoniens, il suffit de
transformer les fonctions hamiltoniennes. C’est certainement une des propriétés les
plus importantes du formalisme hamiltonien. On verra de plus qu’il est très facile de
construire des difféomorphismes symplectiques (sous bien des aspects, on peut dire que
le groupe des difféomorphismes symplectiques est un "gros" groupe).
1.1.3.2. Maintenant, si l’on voit plutôt les systèmes hamiltoniens en termes dynamiques,
il est naturel de chercher des propriétés de préservation, c’est-à-dire des objets

22 Introduction
(ensembles, fonctions, mesures, · · ·) qui sont invariants par le flot. On notera LXH la
dérivée de Lie par rapport au champ de vecteurs XH.
La première propriété de préservation, à l’origine même des systèmes hamiltoniens,
est la préservation de l’énergie.
Proposition 1.6 (Préservation de l’énergie). On a LXHH = 0.
En d’autres termes, la fonction H est constante le long du flot de XH. Chaque orbite
est donc contenue dans un niveau d’énergie H = c, qui est une sous-variété de dimension
2n − 1, si c est une valeur régulière du hamiltonien. De manière infinitésimale, cela se
traduit par le fait que le champ de vecteurs reste toujours tangent au niveau d’énergie.
Cette propriété permet donc de réduire la dimension de l’espace des phases, en particulier
pour un hamiltonien à un degré de liberté (l’espace des phases est de dimension deux),
les orbites sont implicitement définies par l’équation H(q(t), p(t)) = c.
La seconde propriété de préservation, la plus importante du point de vue dynamique,
est la préservation de la forme symplectique Ω.
Proposition 1.7 (Préservation de la forme symplectique). On a LXHΩ = 0.
Autrement dit Ω est constante le long des orbites de XH. Il en est donc de même
pour la forme volume Ω n = Ω ∧ · · · ∧ Ω, et pour sa restriction à un niveau d’énergie.
Si le niveau d’énergie est de volume fini (par exemple dans le cas compact), on peut
appliquer le théorème de récurrence de Poincaré et en déduire que presque toute orbite
est récurrente. La situation diffère donc radicalement des champs de gradients associés
à une métrique riemannienne, où il est facile de voir que la dynamique non triviale
se concentre sur les points fixes (solutions d’équilibres). Les systèmes hamiltoniens sont
donc des systèmes conservatifs particuliers et la récurrence des orbites est un phénomène
typique.
1.1.3.3. Pour conclure, notons que l’on sait dire peu de choses sur les systèmes hamiltoniens
en toute généralité. On va alors adopter la stratégie suivante. On va commencer par
étudier une classe particulière de systèmes hamiltoniens, les systèmes intégrables, que
l’on comprend parfaitement et on va ensuite s’intéresser aux perturbations de systèmes
hamiltoniens intégrables, que l’on appelle systèmes presque-intégrables.
Cette démarche peut se justifier de la manière suivante. D’une part, d’un point de vue
"mathématique", puisque le problème général est difficile il est naturel de commencer
par exhiber une classe de systèmes simples puis d’étudier ce qui se passe "autour".
D’autre part, bien que les systèmes intégrables soient "exceptionnels", un grand nombre
de problèmes "physiques" se ramènent à une perturbation d’un système intégrable, en
particulier c’est le cas du système solaire.
1.2 Systèmes hamiltoniens intégrables
Littéralement, les systèmes hamiltoniens intégrables sont ceux que l’on sait "intégrer",
c’est-à-dire qu’ils possèdent "suffisamment" d’intégrales premières pour que l’on
puisse déterminer "explicitement" leurs solutions.

1.2 - Systèmes hamiltoniens intégrables 23
1.2.1 Structure des systèmes intégrables
1.2.1.1. Donnons nous une variété symplectique (M, Ω) de dimension 2n et une fonction
lisse H : M → R. On commence par rappeler la définition suivante.
Définition 1.8. On dit qu’une fonction lisse f : M → R est une intégrale première du
champ hamiltonien XH si f est constante le long des orbites de XH.
En d’autres termes, la dérivée LXHf est nulle, ou bien encore f commute au sens de
Poisson avec H, c’est-à-dire {f, H} = 0. La conservation de l’énergie se traduit ici par le
fait que les systèmes hamiltoniens possèdent toujours au moins une intégrale première,
à savoir le hamiltonien H lui-même.
1.2.1.2. On peut maintenant définir les systèmes intégrables (au sens de Liouville) et
préciser ce que l’on entend par posséder "suffisamment" d’intégrales premières.
Définition 1.9. On dit que le hamiltonien H est intégrable au sens de Liouville s’il
possède n intégrales premières F = (f1, . . .,fn) indépendantes et en involution, c’est-àdire
:
- {fi, H} = 0, pour i ∈ {1, . . .,n};
- {fi, fj} = 0, pour i, j ∈ {1, . . ., n};
- F : M → R n est une submersion (i.e est partout de rang n).
Commentons un peu la définition. Premièrement, notons que la définition ne demande
que n intégrales premières (on aurait pu choisir fi = H pour un certain i dans
{1, . . ., n}) : c’est une des spécificités des systèmes hamiltoniens, car un champ de vecteurs
arbitraire sur une variété de dimension 2n nécessiterait a priori 2n − 1 intégrales
premières pour pouvoir être intégrable.
Ensuite, on a demandé que l’application F soit une submersion sur M tout entier,
ce qui est une hypothèse très forte et rarement réalisée. Généralement, on se contente
d’imposer que F soit une submersion sur un "gros" ouvert O de M, où par "gros" on
entend par exemple dense ou dont le complémentaire est de codimension positive. On a
donc choisit une définition simplifiée ici, le lieu singulier M \ O pouvant être la source
de fortes complications (y compris dynamiques) que l’on va ignorer.
Enfin, on voit immédiatement d’après les hypothèses que F définit un feuilletage sur
M dont les feuilles (qui sont les composantes connexes des fibres F −1 (c), c ∈ R n ) sont
des sous-variétés invariantes par le flot. Elles sont de plus lagrangiennes, c’est-à-dire
que la restriction de la forme symplectique à ces sous-variétés s’annule et qu’elles sont
de dimension maximale pour cette propriété (à savoir n, la moitié de la dimension de
l’espace des phases). Réciproquement, si une application F = (f1, . . .,fn) : M → R n
définie un feuilletage en sous-variétés invariantes lagrangiennes, alors les composantes
de F sont indépendantes et en involution, autrement dit le système est intégrable au
sens de Liouville.
1.2.1.3. Le théorème principal sur la structure des systèmes hamiltoniens intégrables
est le suivant.
Théorème 1.10. Soit (H, F) un système intégrable au sens de Liouville et c ∈ R n telle
que N = F −1 (c) soit compacte connexe. Alors :

24 Introduction
(i) N est un tore plongé;
(ii) il existe un voisinage ouvert U de N dans M, un voisinage ouvert B de 0 dans
R n et un difféomorphisme Ψ : T n × B → U tels que
- Ψ(T n × {0}) = N ;
- Ψ ∗ Ω|U = Ω0, où Ω0 = dθ ∧ dI, (θ, I) ∈ T n × B ;
- H ◦ Ψ ne dépend que des variables I ∈ B, autrement dit il existe h : B → R
telle que H ◦ Ψ(θ, I) = h(I), pour tout (θ, I) ∈ T n × B.
Les coordonnées (θ, I) ∈ T n ×B sont appelées "action-angle" (les angles sont θ et les
actions I, on devrait donc dire "angle-action"). Ce résultat est généralement attribué à
Liouville, Arnold, Jost et parfois Mineur, ou seulement une combinaison partielle de ces
quatre auteurs (qui varie beaucoup selon le pays dans lequel on se trouve). On pourra
trouver une preuve dans [AA89], [HZ94] ou [Dui80].
Ajoutons quelques remarques sur les hypothèses du théorème précédent.
Si on ne suppose pas la compacité de la feuille N = F −1 (c), il n’est pas difficile de
voir que celle-ci est alors difféomorphe à un produit T n−k × R k , 0 ≤ k ≤ n.
De plus, il existe des généralisations du théorème pour les systèmes proprement
dégénérés (ou super-intégrables, ou non-commutativement intégrables) qui possèdent
m > n intégrables premières (et qui ne sont donc plus toutes en involution), voir par
exemple [Nek72] et [MF78].
Enfin, dans le cas particulier des hamiltoniens de Tonelli (définis sur les cotangents
M = T ∗ X, où X est une variété compacte, et qui vérifient les hypothèses de convexité
usuelles de la théorie de Mather), un résultat récent de Sorrentino ([Sor09]) montre qu’il
est possible d’enlever complètement l’hypothèse d’involution sur les intégrales premières
tout en conservant des informations intéressantes : plus précisément, il montre que pour
toute classe de cohomologie c ∈ H 1 (X, R), le système possède un unique graphe lagrangien
invariant Λc de classe c sur lequel le flot est récurrent, et si X = T n , alors chaque
Λc est un tore sur lequel le flot est linéaire.
1.2.1.4. Désormais, on ne considèrera (presque exclusivement) que des systèmes hamiltoniens
définis sur le produit T n ×B, où B est une boule de R n centrée en 0 (B pouvant
être égale à R n ), muni de coordonnées action-angle (θ, I) et de la forme symplectique
canonique Ω0 = dθ ∧ dI. Dans ce contexte, on s’autorise la définition abusive suivante.
Définition 1.11. On dit qu’un système hamiltonien H est intégrable s’il ne dépend que
des variables d’action I ∈ B, c’est-à-dire s’il existe h : B → R telle que
pour tout (θ, I) ∈ T n × B.
H(θ, I) = h(I),
Ainsi, par le théorème de structure précédent, un tel système intégrable est le modèle
local des systèmes intégrables au sens de Liouville sur une variété symplectique
quelconque. Bien entendu, une meilleure terminologie serait "trivialement intégrable"
ou "intégrable en coordonnées action-angle".
Les solutions des équations associées à un système intégrable sont globales et expli-

1.2 - Systèmes hamiltoniens intégrables 25
cites : elles s’écrivent ˙θ = ∇h(I)
˙
I = 0
et, pour la condition initiale (θ0, I0), elles s’intègrent en
θ(t) = θ0 + t∇h(I0) [Z n ]
I(t) = I0.
Ainsi les variables d’action sont constantes le long du mouvement, ce qui se traduit
géométriquement par le fait que l’espace des phases T n × B se décompose trivialement
en tores lagrangiens invariants T0 = T n × {I0}, paramétrés par les variables d’action
I0 ∈ B, et sur lesquels le champ de vecteurs est constant égal à ω0 = ∇(I0).
1.2.1.5. Avant d’expliquer la dynamique relativement simple des systèmes intégrables,
donnons l’exemple du problème à 2 corps. Considérons deux particules q0 et q1, de masses
m0 et m1, dont les équations sont
q1 − q0
q0 − q1
¨q0 = Gm1
||q1 − q0|| 3, ¨q1 = Gm0
||q0 − q1|| 3.
On se ramène à un problème de Kepler, c’est-à-dire au cas où l’un des deux corps est
fixe, de la manière suivante. Il n’est pas difficile de voir qu’en notant r = q1 − q0 la
position relative des deux corps et s = m0q0 + m1q1 le centre de masse, les équations
précédentes se réduisent aux équations
¨s = 0, ¨r = −µ r
||r|| 3,
(5)
où µ = G(m0 + m1). La première équation de (5) signifie que le centre de masse a une
accélération nulle, cela correspond au principe d’inertie. En plaçant le centre de masse
à l’origine, on obtient
q0 = − m1
r, q1 =
m0 + m1
m0
r.
m0 + m1
Il suffit donc d’étudier l’évolution de r, c’est-à-dire de résoudre la seconde équation
de (5) qui correspond à l’évolution d’une particule de position r subissant l’attraction
d’un corps fixe. Le hamiltonien s’écrit
H = 1
2 ||˙r||2 − µ
r .
Ce dernier est intégrable (au sens de Liouville) : outre le hamiltonien, le moment cinétique
r ∧ ˙r est préservé, ainsi qu’une autre intégrale (appelée intégrale de Laplace).
On peut vérifier de plus que ces intégrales premières sont indépendantes. Si l’espace des
configurations est de dimension 3, on dispose de 5 intégrales premières indépendantes
(alors qu’il en suffit de trois). On peut montrer que les orbites sont des coniques, et
qu’aux énergies négatives ce sont des ellipses, que l’on peut complètement déterminer
par leur demi-grand axe, leur excentricité ainsi que les trois angles d’Euler qui repèrent
le demi-grand axe dans l’espace.
On n’expliquera pas la construction des variables action-angle dans ce contexte,
qui est due à Delaunay (voir [Che89] ou [CC07]). On va se contenter de mentionner
que le hamiltonien écrit en action-angle est proprement dégénéré : il est de la forme
h(I) = h(I1), pour I = (I1, I2, I3), et la variable I1 ne dépend que de la valeur du
demi-grand axe.

26 Introduction
1.2.2 Dynamique des systèmes intégrables
1.2.2.1. Pour comprendre la dynamique d’un système intégrable, on est en particulier
amené à étudier la dynamique d’un champ de vecteurs constant égal à ω ∈ R n sur le
tore, qui engendre bien sûr un flot linéaire (ou un flot de translation de vecteur ω). De
tels flots (ou tores) sont parfois appelés flots (ou tores) de Kronecker de fréquence ω, les
solutions sont qualifiées de quasi-périodiques. Leur dynamique est entièrement comprise,
elle dépend d’une propriété algébrique du vecteur fréquence ω. Plus précisément, à un
tel vecteur on associe un module de résonance
M(ω) = {k ∈ Z n | k.ω = 0} = ω ⊥ ∩ Z n ,
où le . désigne le produit scalaire euclidien de R n , et ⊥ l’orthogonal relativement à ce
produit. C’est un sous-module de Z n , et son rang dicte la dynamique.
Si le rang est nul, autrement dit si
k ∈ Z n , k.ω = 0 =⇒ k = 0,
on dit que la fréquence est non résonante (et par extension le tore est dit non-résonant).
Dans ce cas, il n’est pas difficile de montrer que le flot est minimal (toutes les orbites sont
denses) et uniquement ergodique (il existe une unique mesure de probabilité invariante
par le flot, qui dans ce cas n’est rien autre que la mesure de Haar), et c’est le prototype
de système dynamique à spectre discret.
Si le rang de M(ω) vaut m ≥ 1, la fréquence est dite résonante de multiplicité m, et
dans ce cas le tore T se décompose en une famille continue à m paramètres de sous-tores
invariants, chacun de dimension n − m, sur lesquels le flot est à nouveau minimal et
uniquement ergodique. Par exemple, si m = n, on obtient trivialement un tore invariant
constitué de points fixes et si m = n − 1, le tore est feuilleté en orbites périodiques de
même période.
1.2.2.2. Pour conclure, mentionnons le cas particulier des résonances standard de multiplicité
m, qui par définition sont de la forme (ˆω, 0) ∈ R n−m × R m , avec ˆω ∈ R n−m
non résonant. Le module de résonance de (ˆω, 0) est alors engendré par les m derniers
vecteurs de la base canonique de Z n , autrement dit
M(ˆω, 0) = {k ∈ Z n | k1 = · · · = kn−m = 0}.
Le terme "standard" est justifié par la proposition suivante.
Proposition 1.12. Soit ω une fréquence résonante de multiplicité m. Alors il existe
une matrice A ∈ GLn(Z) telle que
où ˆω ∈ R n−m est non résonant.
ω = A(ˆω, 0),
Ce résultat est très utile lorsque l’on étudie une résonance fixée : en effet A préserve
Z n , elle induit alors un automorphisme linéaire du tore T n qui se relève en un difféomorphisme
linéaire symplectique ΦA(θ, I) = (Aθ, t A −1 I), et on se ramène ainsi à une
résonance standard, quitte à considérer le hamiltonien H ◦ ΦA.

1.3 - Théorie classique des perturbations 27
La preuve de la proposition précédente est une simple conséquence du théorème
de structure des modules de type fini sur les anneaux principaux, plus précisément du
théorème suivant (dit "de la base adaptée", voir [Lan02]) : on peut trouver une base
(f1, . . .,fn) de Zn et des éléments d1, . . ., dm de Z, avec les divisibilités d1| . . . |dm, tels que
(d1f1, . . .,dmfm) soit une base de M(ω). Il suffit alors de choisir A comme la transposée
de la matrice de passage de la base canonique (e1, . . .,en) vers la base (f1, . . .,fn). On
peut également raisonner en termes matriciels : si l’on choisit une base de vecteurs
entiers (k1, . . .,km) de M(ω), on lui associe la matrice
⎛ ⎞
⎜
K = ⎝ .
k 1 1 · · · k n 1
.
k 1 m · · · kn m
⎟
⎠ ∈ Mm,n(Z),
où ki = (k1 i , . . .,kn i ), 1 ≤ i ≤ m. Par le théorème de la base adaptée, cette dernière est
alors équivalente à la matrice diagonale
⎛
d1
⎜
∆ = ⎝
. ..
0 0 · · ·
.
. ..
⎞
0
⎟
. ⎠ ∈ Mm,n(Z),
0 dm 0 · · · 0
plus précisément, K = B∆A pour B ∈ GL(m, Z) et la matrice A ∈ GL(n, Z) convient.
1.3 Théorie classique des perturbations
1.3.0.1. On s’intéresse désormais aux systèmes hamiltoniens presque-intégrables, c’està-dire
aux perturbations de systèmes hamiltoniens intégrables. Ils sont définis par des
hamiltoniens de la forme
H(θ, I) = h(I) + f(θ, I)
(∗)
|f|D < ε

28 Introduction
D’un autre point de vue, les variables d’action I(t) des systèmes intégrables sont
constantes pour tout temps.
Problème 3. Étudier l’évolution des variables d’action I(t) des solutions du système
(∗).
1.3.0.3. Ce dernier problème est motivé par des questions de mécanique céleste, en
particulier par le problème planétaire. Ici on considère un problème à 1 + N corps,
typiquement le Soleil et des planètes du système solaire, de position q0, q1, . . ., qN et de
masse m0, m1, . . .,mN. On suppose que la masse du premier corps est beaucoup plus
grosse que la masse des autres corps, et on note
ε = max
1≤i≤N
mi
Dans le cas du système solaire, ε est de l’ordre de 10 −3 . En première approximation, on
peut donc négliger l’interaction des planètes entre elles pour ne considérer que l’attraction
du corps massif. Cela revient à prendre ε = 0 et le système se découple alors en un
produit de N problèmes de Kepler que l’on sait intégrer. En particulier, les orbites planétaires
sont des ellipses, et le hamiltonien écrit en coordonnées action-angle ne dépend
que des demi-grands axes. Lorsque l’on restaure l’interaction des planètes, le système
planétaire devient une perturbation, de taille environ 10 −3 , d’un système intégrable et
étudier la stabilité des variables d’action revient à étudier la "déformation" éventuelle
des trajectoires elliptiques.
Dans la section suivante on exposera les deux théorèmes fondamentaux qui donnent
des éléments de réponses aux questions précédentes, le théorème des tores invariants
(théorème KAM) et le théorème de Nekhoroshev, mais avant cela on va commencer par
expliquer le principe commun sur lesquels ils reposent, qui est la construction de formes
normales. D’excellentes références sur ce sujet sont [LM88], [Ben05], [Ber06] et bien sûr
[AKN06].
1.3.1 Principe de moyennisation
Le principe de moyennisation est un principe physique qui remonte aux travaux de
Gauss, Lagrange et Laplace en mécanique céleste.
D’un point de vue moderne, on peut l’énoncer ainsi : on introduit la moyenne spatiale
de la perturbation
〈f〉(I) = f(θ, I)dθ,
T n
et on définit le système moyenné 〈H〉 = h + 〈f〉, qui est évidemment intégrable. Alors
le principe de moyennisation consiste à supposer que les solutions du système moyenné
sont de "bonnes approximations" de celles du système complet H = h + f.
L’idée, très naïve, qui sous-tend le principe est la suivante : on écrit le développement
en série de Fourier
f(θ, I) = 〈f〉(I) +
ˆfk(I)e 2πik.θ ,
m0
k∈Z n \{0}
.

1.3 - Théorie classique des perturbations 29
alors les termes ˆ fk(I)e 2πik.θ , pour k ∈ Z n \ {0}, représentent des oscillations qui se
superposent et peuvent donc être négligées.
En étant un peu moins naïf, on voit que ce type de raisonnement n’est valable
que si la dynamique sur chaque tore invariant du système intégrable est suffisamment
"équirépartie", ce qui n’est bien sûr pas toujours le cas. Dans la section suivante, on va
expliquer les problèmes qui se posent en essayant de justifier rigoureusement ce principe.
1.3.2 Théorie des formes normales
1.3.2.1. De manière générale, mais un peu vague, la théorie des formes normales en
systèmes dynamiques consiste à conjuguer (localement ou globalement) le système à un
système "plus simple", autrement dit on cherche à construire des coordonnées simplifiant
le problème.
Pour un système hamiltonien presque-intégrable, le but est bien entendu de le conjuguer
à un système le plus intégrable possible, et d’après la section précédente, le candidat
idéal serait le système moyenné 〈H〉, qui est intégrable. Il faut également demander que
la conjugaison soit symplectique, afin de conserver la structure des équations, et qu’elle
soit ε-proche de l’identité, afin que les solutions du système transformé restent ε-proches
de celles du système initial.
1.3.2.2. Il faut donc commencer par expliquer comment on peut construire de telles
transformations. Deux procédés sont couramment utilisés. La première méthode est
celle des fonctions génératrices, que l’on ne va pas expliquer car on ne va pas l’utiliser.
On utilisera plutôt un second procédé, appelé méthode de Lie, qui a l’avantage de
donner des calculs plus agréables (elle présente aussi l’intérêt de se généraliser aussitôt
en dimension infinie). Sous sa forme la plus simple, la méthode de Lie consiste à choisir
pour transformation Φ le temps 1 du flot d’un champ de vecteurs hamiltonien associé à
une fonction auxiliaire χ, c’est-à-dire Φ = Φ χ
1. Un tel difféomorphisme est évidemment
symplectique (et même exact-symplectique si la variété est exacte), et il est ε-proche
de l’identité si la fonction χ est de taille ε (de manière équivalente on peut également
prendre le temps ε du flot si la fonction est de taille 1).
1.3.2.3. Revenons maintenant à notre problème de forme normale. En pratique, pour
tenter de résoudre l’équation H ◦ Φ = 〈H〉, on utilise un schéma itératif pour éliminer
la perturbation "ordre par ordre" en ε. À l’étape 1, on cherche Φ χ1 tel que
H1 = H ◦ Φ χ1 = 〈H〉 + O(ε 2 ),
c’est-à-dire qu’on cherche à éliminer les angles à l’ordre ε. Ensuite, à l’étape n, pour
n ≥ 2, on a Hn−1 = 〈H〉 + O(ε n ) et on cherche Φ χn tel que
Hn = Hn−1 ◦ Φ χn = 〈H〉 + O(ε n+1 ),
c’est-à-dire qu’on cherche à éliminer les angles à l’ordre ε n . Si le produit infini
Φ = Φ χ1 ◦ Φ χ2 ◦ · · · ◦ Φ χn ◦ · · ·
converge, alors on obtient bien H ◦ Φ = 〈H〉.

30 Introduction
1.3.2.4. Détaillons maintenant une étape de ce schéma. On se donne donc une fonction
χ : T n × B → R de taille ε, on va calculer H ◦ Φ χ afin de voir comment choisir χ pour
se rapprocher du système moyenné. Pour cela, à l’aide de la formule générale
d
(K ◦ Φχt
) = {K, χ} ◦ Φ
dt χ
t
et du théorème de Taylor avec reste intégral, on obtient
H ◦ Φ χ
1 = h ◦ Φ χ
1 + f ◦ Φ χ
1
= h + {h, χ} +
1
= h + f + {h, χ} +
0
(1 − t){{h, χ}, χ} ◦ Φ χ
t dt + f +
1
= 〈H〉 + f − 〈f〉 + {h, χ} +
0
1
{(1 − t){h, χ} + f, χ} ◦ Φ χ
t dt
1
0
0
{f, χ} ◦ Φ χ
t dt
{(1 − t){h, χ} + f, χ} ◦ Φ χ
t dt
où le terme d’ordre ε à éliminer est donné par f − 〈f〉 + {h, χ}, et le reste intégral est
d’ordre ε 2 .
Il faudrait donc choisir χ telle que
{h, χ} + f − 〈f〉 = 0.
C’est l’équation centrale de la théorie des perturbations des systèmes hamiltoniens,
on parle souvent d’équation homologique ou d’équation linéarisée (car elle s’obtient en
linéarisant l’équation de conjugaison de H à 〈H〉). Notons que par la méthode des
fonctions génératrices on aurait trouvé la même équation (au signe près), cette dernière
ne donne donc pas de meilleur résultat. L’équation homologique s’écrit encore
∇h(I).∂θχ(θ, I) = f(θ, I) − 〈f〉(I),
et à une action fixée I, donc à fréquence fixée ω = ∇h(I), on obtient tout simplement
une équation aux dérivées partielles du premier ordre, linéaire et à coefficients constants,
sur le tore T n :
ω.∂θχ = g,
où l’on a noté g = f − 〈f〉. Ce type d’équation se prête bien à l’analyse de Fourier. Dans
notre cas si on écrit les développements en série
g(θ) =
ˆgke 2iπk.θ , χ(θ) =
ˆχke 2iπk.θ ,
k∈Z n
alors ˆg0 = 0 puisque 〈f〉 = ˆ f0 (rappelons que l’on a fixé la variable I pour simplifier la
présentation). On voit donc très facilement que la solution χ est formellement donnée
par ˆχ0 = 0
k∈Z n
ˆχk = ˆ fk
2iπk.ω , k ∈ Zn \ {0}.
1.3.2.5. Notons Ω = ∇h(B) l’espace des fréquences. Pour ω ∈ Ω, à cause de la présence
du produit scalaire k.ω au dénominateur, des problèmes fondamentaux, mis en évidence
par Poincaré, se posent immédiatement.

1.3 - Théorie classique des perturbations 31
Problème des résonances
Si ω est résonante, le produit k.ω s’annule pour un certain multi-entier k non nul,
et ainsi l’équation homologique ne possède tout simplement pas de solution formelle. Il
fallait s’y attendre : dans ce cas le flot linéaire de fréquence ω n’est pas ergodique et
l’approximation de la perturbation par sa moyenne spatiale n’a tout simplement aucun
sens.
De plus, sous des conditions générales sur h, l’ensemble des fréquences résonantes est
dense dans Ω, ainsi pour une perturbation générique l’équation homologique ne possède
de solutions sur aucun ouvert de T n × B : on peut dire que c’est à cause des résonances
qu’une perturbation d’un système intégrable n’est "généralement" pas intégrable (voir
[Koz96] pour plus de détails sur la non-intégrabilité).
Problème des petits diviseurs
Supposons maintenant la fréquence non résonante. On a alors l’existence d’une solution
formelle, mais rien ne garantit la convergence de la solution. En effet, même si
k.ω est non nul pour tout k ∈ Z n \ {0}, le produit scalaire peut (et va) devenir arbitrairement
petit pour des multi-entiers de longueurs arbitrairement grandes, impliquant la
divergence de la série. C’est le fameux phénomène des petits diviseurs.
Enfin on peut également mentionner un troisième problème, qui n’est pas lié aux
résonances ou petits diviseurs, mais qui est incontournable.
Problème des grands multiplicateurs
Il s’agit tout simplement du problème de la convergence du schéma itératif. En
admettant que l’on sache faire face aux problèmes des résonances et petits diviseurs, on
peut alors trouver un changement de variables Φ χ1 qui élimine la perturbation à l’ordre
ε, puis Φ χ2 qui élimine la perturbation à l’ordre ε 2 et ainsi de suite, mais il reste à
montrer la convergence du produit infini
Φ = Φ χ1 ◦ Φ χ2 ◦ · · · ◦ Φ χn ◦ · · ·
et c’est une question délicate. Le terme "grands multiplicateurs" est dû à Poincaré. En
fait, le problème précédent peut se ramener à la convergence d’une série formelle, où le
terme général an se trouve être de l’ordre de A n (n!) α , avec A, α > 0 et qui généralement
diverge. Selon Poincaré, ce sont les séries convergentes au sens des astronomes mais
divergentes au sens des géomètres, en termes modernes ce sont les séries asymptotiques
(voir [Sau95]).
Dans la section suivante, on va donc expliquer comment faire face à ce genre de
problèmes.

32 Introduction
1.4 Théorèmes de stabilité
On va maintenant exposer assez brièvement les deux résultats fondamentaux de
stabilité, le théorème KAM et le théorème de Nekhoroshev.
Le premier résultat affirme que, sous certaines hypothèses, il existe beaucoup de
solutions quasi-périodiques dans les systèmes presque-intégrables. On peut le voir comme
un résultat de stabilité probabiliste : si l’on choisit une solution au hasard, alors avec
une grande probabilité, qui de plus tend vers 1 lorsque ε tend vers zéro, elle est quasipériodique
et on peut en déduire facilement que les variables d’action ont une variation
d’ordre ε pour tout temps.
Le second résultat concerne toutes les solutions, mais fournit bien sûr des résultats
moins précis. Sous certaines conditions, les variables d’action de toute solution sont
presque constantes pendant un intervalle de temps exponentiellement long par rapport
à l’inverse de la taille de la perturbation. On peut le voir comme un résultat de stabilité
"effective", plus "physique" que le précédent.
1.4.1 Théorie KAM
1.4.1.1. A l’origine, la théorie KAM, d’après Kolmogorov, Arnold et Moser, étudie la
persistance de certaines solutions quasi-périodiques (autrement dit de tores invariants
quasi-périodiques) pour des perturbations de systèmes hamiltoniens intégrables. On renvoie
à [Bos86], [Pös01] ou [dlL01] pour de très bonnes introductions. Dans un sens plus
large, on parle de théorie KAM (ou de méthodes KAM) lorsque se présente un problème
de petits diviseurs en dynamique (linéarisation des difféomorphismes du cercle,
des germes de fonctions holomorphes au voisinage d’un point fixe indifférent, réductibilité
des cocycles quasi-périodiques proches d’un cocyle constant, ...), ou dans d’autres
domaines des mathématiques (géométrie, équations aux dérivées partielles).
1.4.1.2. Considérons donc un système presque-intégrable de la forme
H(θ, I) = h(I) + f(θ, I)
|f|D < ε 0 et τ ≥ 0, un vecteur ω ∈ Rn est dit (γ, τ)-diophantien
s’il vérifie l’inégalité
|k.ω|1 ≥ γ|k| −τ
1 ,
pour tout k ∈ Z n \ {0}, où | . |1 désigne la norme ℓ 1 .

1.4 - Théorèmes de stabilité 33
et
On note D τ γ
l’ensemble de ces vecteurs, et on pose
Dγ =
D τ γ , Dτ =
τ>0
D =
τ>0,γ>0
est l’ensemble des vecteurs diophantiens. Un tel vecteur ω ∈ D est donc "loin" des
résonances. Lorsque |k|1 devient grand on peut alors contrôler la petitesse des diviseurs
|k.ω|. Pour que D τ soit non vide, on montre facilement qu’il est nécessaire (et suffisant)
que τ ≥ n −1. Si τ > n −1, il n’est alors pas difficile de prouver que D τ un ensemble de
mesure de Lebesgue totale. En revanche, si τ = n −1, il est plus difficile de montrer que
la mesure de Lebesgue est nulle et que la dimension de Hausdorff est maximale. Dans
tous les cas, c’est un ensemble maigre au sens de la catégorie de Baire.
1.4.1.3. Le premier résultat de persistance de tores invariants, contenant les principes
fondamentaux de la théorie, est dû à Kolmogorov ([Kol54]). On dit que la partie intégrable
h : B → R est non dégénérée au sens de Kolmogorov si en tout point l’application
fréquence ∇h est un difféomorphisme local. On note Ω = ∇h(B) l’espace des fréquences,
et
Ωγ = {ω ∈ Ω ∩ Dγ, d(ω, ∂Ω) ≥ γ}
un espace de fréquences diophantiennes suffisamment loin du bord de Ω. On peut alors
énoncer de manière informelle le résultat suivant.
Théorème 1.14 (Kolmogorov). On considère un système hamiltonien (∗), et on suppose
que :
(i) le système est analytique;
(ii) h est non dégénérée au sens de Kolmogorov.
Alors il existe une constante c qui ne dépend que de h telle que pour ε < cγ 2 , au voisinage
de toute solution quasi-périodique du système non perturbé, de fréquence ω ∈ Ωγ, le
système possède une solution quasi-périodique de même fréquence.
En termes plus précis, la dernière phrase signifie que pour tout ω ∈ Ωγ, il existe un
plongement
Ψ ε ω : T n → T n × B,
dont l’image T est invariante par le champ hamiltonien, et qui conjugue la restriction
du champ à T à un flot linéaire sur T n de fréquence ω. De plus, si l’on suppose (sans
perte de généralités) que le tore non perturbé de fréquence ω est situé en I = 0, alors
Ψ ε ω est √ ε-proche du plongement standard Ψ 0 ω qui envoie Tn sur T n × {0}.
Pour une exposition détaillée de la preuve de Kolmogorov, on pourra consulter
[BGGS84].
D τ γ
γ>0
D τ γ ,
1.4.1.4. Les deux idées essentielles de la preuve sont les suivantes.
La première idée, que l’on a déjà mentionnée, est de prescrire la fréquence initiale ω
du tore recherché et de la choisir diophantienne. Dans l’approche de Kolmogorov, cela se
traduit par la construction d’une conjugaison Φ du système à la forme normale suivante
K(θ, I) = ω.I + O(|I| 2 )

34 Introduction
dite "de Kolmogorov" qui possède, en I = 0, un tore invariant de fréquence ω. Le
plongement Ψ ε ω est alors donné par Ψ ε ω = Φ ◦ Ψ 0 ω. Pour construire cette conjugaison,
on peut procéder de manière itérative comme on l’a expliqué précédemment. À chaque
étape, quitte à effectuer de petites translations dans l’espace des fréquences (ce qui est
possible par la non-dégénérescence de h), on obtient une équation homologique que l’on
sait résoudre grâce à l’hypothèse arithmétique sur la fréquence ω.
Mais il reste le problème de la convergence du schéma itératif (les grands multiplicateurs),
et c’est ici qu’intervient la seconde idée de Kolmogorov. Au lieu d’utiliser un
schéma que l’on pourrait qualifier de linéaire, comme en théorie classique des perturbations,
il utilise une méthode de Newton. Il s’agit d’un schéma de type quadratique :
au bout de n étapes, au lieu d’avoir une erreur de l’ordre de εn+1 , elle est de l’ordre
ε2n. Ceci permet de compenser les mauvaises estimations liées aux petits diviseurs et de
garantir la convergence.
1.4.1.5. Une preuve techniquement différente mais reposant sur les mêmes idées a
été donnée ensuite par Arnold ([Arn63a]), et un théorème de préservation de courbes
invariantes pour les perturbations de certains difféomorphismes intégrables de l’anneau
A (en différentiabilité finie) a été démontré par Moser ([Mos62]), d’où l’acronyme KAM,
pour Kolmogorov, Arnold et Moser.
Pour une preuve plus moderne, on renvoie à [Pös01] pour ce qui est communément
appelé la théorie KAM "à paramètres" : l’idée est de considérer les fréquences ω = ∇h(I)
comme des paramètres indépendants des variables d’action I pour obtenir une seule
transformation
Φ : T n × Ω −→ T n × B
qui "redresse" tous les tores invariants de fréquences ω ∈ Ωγ. Cela permet d’étudier
plus simplement la question de la régularité des tores en fonction de la fréquence, et
d’obtenir des renseignements plus quantitatifs. En particulier on montre plus facilement
que la mesure relative du complémentaire de l’ensemble des tores invariants est d’ordre
√ ε. Cette démarche est implicite chez Arnold ([Arn63a]) et Moser ([Mos67]), et complètement
explicite chez Pöschel ([Pös82], voir aussi [CG82]).
Il existe également beaucoup d’autres méthodes de preuve qui diffèrent plus ou moins
du schéma classique proposé par Kolmogorov et complété par Arnold. Citons en quelques
unes.
Une première méthode, due essentiellement à Zehnder ([Zeh75], [Zeh76]) et à Herman
([Bos86], [Féj04]) réduit le problème à l’application d’un théorème d’inversion local (dit
de Nash-Moser) dans une certaine échelle d’espaces fonctionnels. Dans cette direction,
on peut également consulter [Féj10] pour une preuve récente et plus simple.
Une autre méthode de preuve, due à Eliasson ([Eli96]), consiste à analyser directement
la convergence des séries classiques de la théorie des perturbations (séries de
Lindstedt). Plus précisément, il s’agit d’ajouter des termes à la série puis de les regrouper
de manière adéquate afin de mettre en évidence certaines compensations de signes
entraînant la convergence de la série.
Une autre approche encore, que l’on doit à Levi-Moser dans le cas des courbes invariantes
([LM01]) et Zehnder-Salamon dans le cas général ([SZ89]), est basée sur l’utilisation
du formalisme lagrangien.

1.4 - Théorèmes de stabilité 35
Signalons également une preuve, due à Khanin, Lopes Dias, et Marklof ([KLDM06]
et [KLDM07]) où l’analyse des petits diviseurs est remplacée par de l’approximation
diophantienne simultanée, plus précisément par un procédé de renormalisation basé sur
un algorithme multi-dimensionnel de fractions continues.
Enfin, une méthode a été proposé récemment par Rüssmann ([Rüs09], d’après l’auteur
ce travail remonte aux années 1980), méthode reprise ensuite par Pöschel ([Pös09])
dans le cadre plus simple d’une perturbation d’un champ de vecteurs constant sur le
tore. La nouveauté réside dans la disparition de toute convergence quadratique (ou plutôt
super-linéaire), qui semblait pourtant essentielle. L’idée principale est de décomposer
la perturbation f en une partie "infrarouge" f≤ et une partie "ultraviolette" f>, mais
à la différence d’une décomposition classique selon l’ordre des coefficients de Fourier,
Rüssmann introduit dans la partie ultraviolette une certaine fraction des coefficients de
Fourier de faible ordre. Il obtient ainsi, après moyennisation, de meilleures estimées sur
la partie infrarouge valables sur de plus grand domaines. L’itération du schéma est alors
très simple, et de manière surprenante, la convergence est très lente (à la i-ème étape,
le nouveau terme d’erreur fi vérifie une inégalité du type |fi| ≤ q i |f|, avec 0 < q < 1
proche de 1).
1.4.1.6. Notons également que toutes les hypothèses du théorème peuvent être affaiblies.
En ce qui concerne la régularité, on sait depuis Moser ([Mos62]) que la théorie est
valable en différentiabilité finie : on a besoin que le système soit de classe C k pour
k > 2n (voir [Pös82], [Sal04], [SZ89] et [Alb07]), et l’on sait que cette hypothèse est
essentiellement optimale pour n = 2 ([Her86], voir aussi [KT09] pour un résultat optimal
dans le cadre voisin de la linéarisation des difféomorphismes du cercle). Le cas où le
système est de classe Gevrey a été traité par Popov ([Pop04]).
Pour ce qui est de l’hypothèse de non-dégénérescence, on peut remplacer la condition
de Kolmogorov par la condition de non-dégénérescence iso-énergétique d’Arnold
([Arn63a]), où l’on demande que l’application fréquence ne s’annule pas et que l’application
"fréquence projective" associée (c’est-à-dire les n −1 rapports de fréquences) soit
un difféomorphisme local en restriction à un niveau d’énergie. La conclusion en est donc
un peu modifiée : les tores KAM ont la même énergie et la même fréquence "projective"
que les tores non perturbés, mais plus nécessairement la même fréquence. Notons
que l’hypothèse de non-dégénérescence de Kolmogorov et de non-dégénérescence d’Arnold
sont complètement indépendantes. Dans [Arn63a], Arnold démontre également un
théorème KAM pour des hamiltoniens intégrables proprement dégénérés, en vue d’une
application au problème planétaire (voir aussi [Féj04]). Pour ce qui est de l’hypothèse la
plus faible, minimale dans le cas analytique, elle est due à Rüssmann ([Rüs01]) : l’image
de l’application fréquence ne doit pas être contenue dans un hyperplan.
Enfin l’hypothèse arithmétique sur la fréquence peut également être affaiblie, il suffit
d’utiliser une condition arithmétique de type Brjuno ([Rüs01]).
1.4.1.7. Pour conclure, notons que la théorie KAM ne s’applique pas aux tores résonants.
Ces derniers sont généralement détruits par la perturbation et donnent naissance
à une dynamique "chaotique". Néanmoins, ces tores ne disparaissent pas totalement. En
effet, considérons un tore m-résonant, qui se décompose en une famille à m paramètres
de tores de dimension n − m, et écrivons sa fréquence ω = (ˆω, 0) ∈ R n−m × R m .

36 Introduction
Pour m = n − 1, un tore résonant est feuilleté en orbites périodiques de même
période. Bernstein et Katok ([BK87]) ont montré que si la partie intégrable est convexe,
il existe au moins n orbites périodiques qui persistent. Pour m = 1, un résultat analogue
est dû à Cheng ([Che99]), il démontre l’existence d’au moins deux tores de dimension
n − 1, sous l’hypothèse que la partie intégrable soit non dégénérée et que la fréquence
restreinte ˆω soit diophantienne.
En revanche, pour les cas intermédiaires 1 < m < n−1, on ne dispose que de résultats
partiels ([Tre91], [CW99]), qui requièrent des hypothèses supplémentaires sur la partie
intégrable ou sur la perturbation. La conjecture est la suivante : dans un système non
dégénéré, pour un tore m-résonant avec une fréquence restreinte ˆω diophantienne, il
subsiste au moins m + 1, et génériquement 2 m , tores de dimension n − m. Autrement
dit, leur nombre devrait être égal au nombre de points fixes d’une fonction régulière sur
le tore T m .
1.4.2 Théorie de Nekhoroshev
1.4.2.1. On peut déduire de la théorie KAM que pour un système presque-intégrable
suffisamment régulier et dont la partie intégrable n’est pas trop dégénérée, il existe une
constante c ′ telle que pour ε assez petit, on ait
|I(t) − I0| ≤ c ′√ ε, t ∈ R,
pour "beaucoup" d’actions initiales I0 ∈ B, celles dont l’évolution est quasi-périodique.
En effet, d’une part on sait que l’ensemble des tores KAM est gros au sens de la mesure
car son complémentaire a une mesure relative d’ordre √ ε (il n’est pas de mesure totale
à cause de la condition ε < cγ 2 ), et d’autre part chaque tore est √ ε-proche du tore non
perturbé sur lequel les variables d’action du système non perturbé sont constantes, ce
qui explique la variation des actions du système perturbé.
Pour n = 2, cette propriété de stabilité est même vraie pour toute solution, dans le
cadre du théorème KAM iso-énergétique d’Arnold : sur chaque niveau d’énergie, qui est
de dimension 3, persiste une famille de tores invariants de dimension 2 telle que chaque
composante connexe du complémentaire est bornée. Alors ou bien la solution est quasipériodique,
et sa variation est d’ordre √ ε, ou bien elle est "coincée" entre deux solutions
quasi-périodiques, et un argument utilisant la mesure des tores préservés montre que sa
variation est également d’ordre √ ε.
En 1964, Arnold ([Arn64]) a démontré qu’une telle propriété ne subsiste pas pour
n ≥ 3. Il a construit un exemple de système hamiltonien à trois degrés de liberté qui
possède une solution (θ(t), I(t)) telle que
|I(τ) − I0| ≥ 1,
avec τ = τ(ε), et ceci pour tout ε > 0 (on donnera plus de détails dans la section
suivante). Donc pour n ≥ 3, la théorie KAM ne fournit pas de résultat de stabilité
valable pour toutes les solutions.
On dit alors qu’un système est effectivement stable si il existe des constantes positives
b, c telles que pour toute action initiale I0,
|I(t) − I0| ≤ cε b , |t| ≤ T(ε),

1.4 - Théorèmes de stabilité 37
avec
lim T(ε) = +∞.
ε→0
Pour n = 2, la théorie KAM nous donne donc des résultats de stabilité perpétuelle
(ou de stabilité en temps infini), c’est-à-dire T(ε) = +∞ (et b = 1/2). On parlera de
stabilité polynomiale (resp. exponentielle, resp. super-exponentielle) si T(ε) est d’ordre
ε−k , k ∈ N∗ (resp. exp(ε−1 ), resp. exp exp(ε−1 )).
1.4.2.2. Dans les années 1970, Nekhoroshev a démontré que, si le système est analytique
et si la partie intégrable vérifie une condition "générique", alors il est exponentiellement
stable. Voici un énoncé informelle.
Théorème 1.15 (Nekhoroshev). On considère un système hamiltonien (∗), et on suppose
que :
(i) le système est analytique;
(ii) h satisfait une condition "générique".
Alors il existe des constantes c1, c2, c3, a, b, ε0 qui ne dépendent que de h telles que pour
ε ≤ ε0,
|I(t) − I0| ≤ c1ε b , |t| ≤ c2 exp c3ε −a ,
pour toute action initiale I0 ∈ BR/2.
Pour la preuve originale, on ne peut consulter que [Nek77], [Nek79].
Les constantes a et b sont appelés exposants de stabilité, la valeur de a est la plus
importante car elle fournit le temps précis de stabilité, il est de l’ordre de exp (ε −a ). La
valeur donnée par Nekhoroshev est a ∼ n 2 , naturellement elle tend vers zéro lorsque n
tend vers l’infini.
L’hypothèse faite sur h par Nekhoroshev est une certaine condition de transversalité
quantitative, que l’on appelle escarpement (ou raideur). Elle semble adaptée à la preuve
et n’est donc pas très conceptuelle. En revanche, on a une caractérisation géométrique
simple due à Ilyashenko et Niederman ([Ily86], [Nie06]) : la restriction de h à tout sousespace
affine propre n’admet que des points critiques isolés. Notons que Nekhoroshev
supposait également que h ne possède pas de points critiques et qu’il soit non dégénérée
au sens de Kolmogorov, mais ces hypothèses sont en fait superflues ([Nie07b]).
On a qualifié la condition sur h de générique : ce n’est pas très clair, à ma connaissance
l’espace des fonctions non escarpées est de codimension infinie dans l’espace des fonctions
de classe C ∞ . Une condition générique d’escarpement diophantien a été donnée par
Niederman dans [Nie07b] : c’est une condition prévalente dans l’espace des fonctions de
classe C k , k > 2n + 2, la prévalence étant une généralisation possible de la notion de
mesure de Lebesgue pleine pour des espaces de dimension infinie.
1.4.2.3. Expliquons maintenant rapidement les principales idées, toujours dans l’optique
des trois difficultés de la théorie classique des perturbations.
Tout d’abord, le problème des résonances ne peut pas être ignoré ici. La stratégie
est alors de changer de système moyenné : on remplace la moyenne spatiale 〈f〉 par la
moyenne temporelle le long du flot non perturbé [f], c’est-à-dire
t
1
[f] = lim f ◦ Φ
t→∞ t
h
sds ,
0

38 Introduction
et on considère le nouveau système moyenné [H] = h + [f]. Ce dernier n’est plus nécessairement
intégrable : pour une action I ∈ B de fréquence ω = ∇h(I), on a
t
1
[f](θ, I) = lim f(θ + sω, I)ds .
t→∞ t
Elle dépend donc de ω, et plus précisément de son module de résonance M(ω). Lorsque
ω est non résonant, alors le flot linéaire de fréquence ω est ergodique et par le théorème
(ergodique) de Birkhoff on retrouve [f] = 〈f〉, mais ce n’est pas vrai dans le cas général. Il
est bon de garder en tête l’exemple des résonances standard : si ω = (ˆω, 0) ∈ R n−m ×R m
avec ˆω non résonant, alors
[f](θ, I) = f(θn−m+1, . . .,θn, I),
et on peut déduire immédiatement que les variables d’action I1, . . .,In−m sont constantes
(mais on ne sait rien de plus sur les variables In−m+1, . . .,In).
Lorsque l’on compare le système H au système moyenné [H], on retrouve le problème
des petits diviseurs, l’idée est alors de considérer des fréquences "résonantesdiophantiennes".
Si l’on pense aux résonances standard (ˆω, 0), cela revient à demander
que ˆω soit diophantienne. Plus précisément, à un sous-module M de Z n fixé (que l’on
doit voir comme un module de résonance), on considère dans l’espace des actions le
domaine "non résonant modulo M", BM, défini de la manière suivante : si I ∈ BM et
ω = ∇h(I), alors ou bien k.ω = 0 pour k ∈ M, ou bien on a un contrôle sur |k.ω|
pour les multi-entiers k /∈ M. C’est sur de tels domaines BM qu’il est alors possible de
comparer H à [H].
Il reste enfin le problème des grands multiplicateurs, à savoir la convergence du
schéma itératif. Il n’y a malheureusement pas de solution ici, en général il y a divergence.
Une astuce consiste alors à faire un nombre fini mais "asymptotiquement infini"
d’étapes : à ε fixé, on fait un nombre d’étapes de l’ordre de ε −a , 0 < a < 1, et donc
lorsque ε tend vers zéro, ce nombre tend vers l’infini. L’effet est le suivant : sur un
domaine non résonant, on peut conjuguer H à [H] + ˜ f, où ˜ f est un terme général exponentiellement
petit en ε −a . On en déduit alors un résultat de stabilité exponentielle
mais seulement "local" et "partiel" : local dans le sens où il n’est valable que pour les
solutions qui restent dans un domaine non résonant où la forme normale est valide, et
partiel puisque comme le système moyenné n’est plus nécessairement intégrable, on ne
peut contrôler l’évolution des variables d’action que dans certaines directions (penser au
cas d’une résonance standard).
Enfin un argument final très compliqué de "géométrie des résonances" permet de
conclure (c’est certainement l’étape la plus difficile). Il consiste à recouvrir l’espace des
phases par des domaines non résonants de manière adéquate, puis à l’aide d’une propriété
de transversalité du système intégrable, on recolle les résultats locaux et partiels de
stabilité valables sur chaque domaine en un résultat de stabilité globale.
1.4.2.4. La preuve précédente est très compliquée, mais elle se simplifie dans le cas particulier
où h est quasi-convexe (c’est-à-dire que ses sous-niveaux d’énergie sont convexes) :
sur des domaines non résonants on obtient un résultat de stabilité locale mais non plus
partielle, ce qui facilite par la suite la géométrie des résonances.
Cependant, dans le cas quasi-convexe, on dispose d’une méthode "révolutionnaire"
due à Lochak, qui est à la fois plus simple, plus élégante et donne des améliorations
0

1.4 - Théorèmes de stabilité 39
quantitatives importantes ([Loc92], voir aussi [Loc93] pour une exposition non technique
des idées). Le point essentiel dans l’approche de Lochak est de ne considérer que les
fréquences périodiques, c’est-à-dire les vecteurs ω ∈ R n pour lesquels il existe un réel
T > 0 tel que
Tω ∈ Z n .
L’exemple le plus simple est donné par un vecteur dont les composantes sont rationnelles.
Ce sont les résonances maximales, dans le sens où le module M(ω) est de rang n − 1
(et réciproquement tout sous-module de Z n de rang n − 1 est de cette forme). Le terme
périodique vient bien sûr du fait qu’un flot linéaire de fréquence ω a toutes ses orbites
T-périodiques.
Pour ce qui est de la construction de formes normales, elle est grandement simplifiée
dans ce cas particulier car il n’y a pas de petits diviseurs. En effet, si k ∈ Z n \ {0} ne
résonne pas avec ω, c’est-à-dire si k.ω = 0, alors
|k.ω| ≥ T −1 ,
et de plus la borne inférieure obtenue est uniforme en k. D’un autre point de vue,
l’équation homologique ω.∂θχ = g se résout facilement par une formule intégrale
χ(θ) = T −1
T
et les petits diviseurs n’apparaissent pas.
0
g(θ + sω)sds,
En utilisant la quasi-convexité, on en déduit facilement un résultat de stabilité locale,
puis à l’aide d’un théorème de Dirichlet sur l’approximation des vecteurs par des vecteurs
à coordonnées rationnelles, on obtient de manière remarquablement simple le résultat
de stabilité globale.
Pour ce qui est des améliorations quantitatives, Lochak et Neishtadt ([LN92]) ont
montré que l’on peut choisir les exposants de stabilité
a = b = 1
2n .
On verra par la suite que la valeur de l’exposant a est quasiment optimale. De plus, l’approche
de Lochak permet de mettre en évidence un phénomène particulier et extrêmement
surprenant de stabilisation par les résonances : si une solution passe suffisamment
proche d’une résonance de multiplicité m fixé, 0 ≤ m ≤ n − 1, alors on a des exposants
de stabilité locaux
am = bm =
1
2(n − m) .
Bien que les résonances soient la cause principale de l’instabilité asymptotique, elles
favorisent la stabilité en temps fini. Ajoutons que tous ces résultats peuvent également se
retrouver par l’approche classique de Nekhoroshev, comme l’a montré Pöschel ([Pös93]).
Notons enfin qu’en utilisant l’approche de Lochak, Marco et Sauzin ([MS02]) ont
généralisé le résultat pour des hamiltoniens de classe α-Gevrey, α ≥ 1, en obtenant des
exposants de stabilité
a = 1 1
, b =
2αn 2n ,

40 Introduction
et des exposants de stabilité locaux
am =
1
2α(n − m) , bm =
1
2(n − m) .
Puisque les fonctions 1-Gevrey sont exactement les fonctions analytiques, on obtient
bien une généralisation des résultats précédents.
1.5 Exemples d’instabilité
Jusqu’à présent, on a vu que les solutions d’un système hamiltonien presqueintégrable
sont stables sur une grande partie de l’espace des phases par le théorème
KAM et que, dans le complémentaire, elles sont stables pendant un grand intervalle de
temps par le théorème de Nekhoroshev. On s’intéresse maintenant à la limite de validité
de ces résultats de stabilité, autrement dit on veut étudier le comportement des
solutions, dans l’espace et le temps qui ne sont pas couverts par la théorie KAM et la
théorie de Nekhoroshev.
Le but de cette section est de décrire le mécanisme d’instabilité "globale" introduit
par Arnold, ainsi que les diverses généralisations qui ont suivi. On insiste sur le fait
qu’une grande partie des travaux récents sur l’instabilité utilisent des méthodes variationnelles
introduites par Mather, mais que par manque de connaissances de l’auteur
dans ce domaine, on va se limiter à la description de certains aspects "géométriques".
Pour une jolie présentation du mécanisme d’Arnold, on pourra consulter [Mar96] ou
[Ber06], et pour plus d’informations, la référence incontournable est [Loc99].
1.5.1 Le mécanisme d’Arnold
1.5.1.1. Le point de départ est l’exemple suivant. Considérons le hamiltonien défini sur
T 3 × R 3 , qui dépend de deux paramètres positifs ε et µ :
Hε,µ(θ, I) = 1
2 (I2 1 + I2 2 ) + I3 + ε(cos 2πθ1 − 1) + εµ(cos 2πθ1 − 1)(cos 2πθ2 + sin 2πθ3).
En 1964, Arnold a démontré le théorème suivant.
Théorème 1.16. Pour tout ε > 0, il existe µ0(ε) tel que pour 0 < µ < µ0(ε), le système
hamiltonien défini par Hε,µ possède une orbite (θ(t), I(t)) vérifiant
pour un temps τ = τ(ε).
|I(τ) − I(0)| ≥ 1,
On appelle généralement "diffusion" d’Arnold ce phénomène d’instabilité, autrement
dit une variation d’ordre 1 des variables d’action pour tout ε > 0, aussi petit soit-il. Le
temps τ(ε) est appelé temps d’instabilité (ou de dérive, ou de diffusion). Le mécanisme
introduit par Arnold pour prouver ce résultat est contenu dans [Arn64] (voir également
[AA89]), et nous allons l’expliquer brièvement.

1.5 - Exemples d’instabilité 41
1.5.1.2. Commençons par le cas ε = µ = 0. Alors
H0,0(θ, I) = 1
2 (I2 1 + I2 2 ) + I3
est un système intégrable quasi-convexe (le plus simple possible), les variables d’action
sont donc constantes.
Maintenant, pour ε > 0, µ = 0, le système
Hε,0(θ, I) = 1
2 (I2 1 + I2 2 ) + I3 + ε(cos 2πθ1 − 1)
reste intégrable au sens de Liouville, et il est facile de voir que les variables d’action
I2, I3 sont constantes tandis que I1 a une variation maximale de 2 √ ε.
La caractéristique la plus importante du système Hε,0 est qu’il possède des objets
invariants hyperboliques. En effet, on voit facilement que c’est le produit direct d’un
pendule (dans les coordonnées (θ1, I1)) et d’un système intégrable en action-angle. Le
pendule possède un point fixe hyperbolique en {θ1 = 0, I1 = 0}, dont les variétés stable
et instable (les séparatrices) coïncident. Ainsi, dans chaque niveau d’énergie, le système
Hε,0 possède une famille continue à un paramètre de tores invariants de dimension 2
partiellement hyperboliques : par exemple au niveau d’énergie zéro on trouve
Ts = {θ1 = 0, I1 = 0, I2 = s, I3 = −s 2 /2}, s ∈ R.
Chacun de ces tores possède des variétés stable et instable de dimension 3 qui coïncident
(elles sont données par le produit direct du tore avec la variété stable et instable du point
fixe hyperbolique). Pour des raisons de dimension, on voit facilement que l’hyperbolicité
des tores est seulement partielle (ils ne sont pas normalement hyperboliques).
Regardons maintenant ce qui peut se passer pour µ > 0. D’une part, on remarque
que dans l’exemple d’Arnold les champs de vecteurs hamiltoniens engendrés par Hε,0 et
Hε,µ coïncident en {θ1 = 0}, ainsi dans chaque niveau d’énergie on a encore une famille
continue de tores partiellement hyperboliques pour Hε,µ.
D’autre part, on peut montrer la propriété suivante : il existe µ0(ε) > 0 tel que pour
0 < µ < µ0(ε), les variétés stable et instable des tores Ts se coupent transversalement
(dans un niveau d’énergie) le long d’au moins une orbite homocline. Dans cet exemple,
cela peut se montrer par un simple calcul d’intégrale (dite de Poincaré-Melnikov) mais
cela nécessite de choisir µ0(ε) exponentiellement petit par rapport à ε.
On obtient alors une famille à un paramètre de tores partiellement hyperboliques
(Ts)s∈R qui possèdent des orbites homoclines transverses, et on peut en déduire que
deux tores consécutifs suffisamment proches sont connectés par une orbite hétérocline
transverse. On peut alors extraire de cette famille une suite finie de tores
Ti = Tsi , 1 ≤ i ≤ N,
avec les propriétés suivantes : pour chaque 1 ≤ i ≤ N, la dynamique sur le tore Ti
est minimale et sa variété instable W +
i coupe transversalement la variété stable W −
i+1
du tore Ti+1 le long d’une orbite hétérocline. C’est ce qu’on appelle une "chaîne de
transition", et en choisissant s1 < 0 et sN > 1, on peut facilement en déduire l’existence
d’une pseudo-orbite qui a une dérive d’ordre 1.

42 Introduction
Il reste enfin à suivre cette pseudo-orbite, c’est-à-dire à construire une orbite qui longe
la chaine de transition. Pour cela, Arnold introduit la notion de "propriété d’obstruction"
que l’on peut énoncer ainsi : Ti possède cette propriété lorsque toute sous-variété M,
invariante par le flot et qui coupe transversalement (en restriction à un niveau d’énergie)
la variété instable W +
−
i , vérifie Wi ⊆ M. En utilisant le fait que les tores qui constituent
la chaîne de transition possèdent cette propriété d’obstruction, un argument topologique
simple permet de conclure à l’existence d’une orbite de dérive.
1.5.1.3. Voici donc le mécanisme proposé par Arnold, tel qu’il l’a illustré sur un exemple
simple mais concret. Il a également conjecturé que ce phénomène d’instabilité est "générique".
Cependant, certains arguments qu’il utilise dans la construction de son exemple
ne sont pas totalement justifiés, de plus, des difficultés apparaissent lorsque l’on tente
d’utiliser ce mécanisme dans des situations plus générales.
Premièrement, la propriété d’obstruction introduite par Arnold n’est pas démontrée,
et en raison de l’hyperbolicité seulement partielle des tores, cela ne résulte pas immédiatement
du lemme d’inclinaison (λ-lemme) classique valable pour les points fixes
hyperboliques. De même, l’existence de connexions hétéroclines transverses entre deux
tores suffisamment proches ne se déduit pas trivialement de l’existence d’orbites homoclines
transverses. Les justifications sont venues plus tard, d’abord [Dou88] et [CG94]
dans des cas particuliers, puis [Mar96] (voir aussi [Cre97]) pour un cas plus général qui
s’applique à l’exemple d’Arnold. On dispose aujourd’hui d’un lemme d’inclinaison suffisamment
général dans ce contexte partiellement hyperbolique (voir [FM00] et [FM01]).
Cependant, on verra dans la suite que le cadre le plus adapté à ce genre de problèmes est
celui des variétés normalement hyperboliques. On pourra également consulter [Bes96]
pour une construction variationnelle d’une orbite de dérive dans l’exemple d’Arnold.
Ensuite, une difficulté plus sérieuse apparaît si l’on cherche à construire des exemples
plus généraux. Rappelons que dans l’exemple d’Arnold, le champ de vecteurs engendré
par la perturbation en µ s’annule le long d’une sous-variété, ce qui permet de continuer
à des valeurs µ > 0 la famille continue de tores partiellement hyperboliques présente
à µ = 0. Pour une perturbation "générique", il n’est pas possible de conserver cette
propriété. Cependant, une version du théorème KAM permet de montrer que beaucoup
de tores partiellement hyperboliques persistent (voir [Gra74] par exemple), mais alors
ils ne forment plus une famille continue : seuls persistent les tores avec une fréquence
suffisamment non résonante. Il va alors se créer des "trous" entre deux tores invariants
consécutifs, si bien que la variété stable d’un tore ne coupe plus nécessairement la variété
instable du tore suivant : c’est le "large gap problem".
Enfin, en vue de la généricité de ce mécanisme, la difficulté la plus sérieuse est la
présence de deux paramètres de perturbation ε et µ que l’on ne peut pas choisir de
la même manière. On voit clairement dans l’exemple que le paramètre ε introduit de
l’hyperbolicité dans le système tout en conservant l’intégrabilité, tandis que le paramètre
µ détruit l’intégrabilité et rend possible l’instabilité. Cependant, il est important de noter
qu’on ne sait montrer l’existence d’orbites de dérive que si µ est choisi exponentiellement
petit par rapport à ε. Autrement dit, il est plus correct de considérer l’exemple comme
une perturbation de taille µ du hamiltonien
Hε,0(θ, I) = H1(θ1, I1) + H2(I2, I3) = 1
+ I3.
2 I2 1 + ε(cos2πθ1 − 1) + 1
2 I2 2
Ainsi le mécanisme d’Arnold ne s’applique vraiment qu’aux perturbations d’un sys-

1.5 - Exemples d’instabilité 43
tème intégrable (au sens de Liouville) partiellement hyperbolique, que l’on appelle "a
priori instable" (suivant la terminologie introduite dans [CG94], on parle également de
système initialement hyperbolique), par opposition aux systèmes intégrables en coordonnées
action-angle où l’hyperbolicité est absente et qui sont "a priori stables" (ou
initialement elliptiques).
1.5.2 Résonances simples et résonances multiples
On va maintenant expliquer comment l’exemple d’Arnold permet d’étudier plus généralement
l’instabilité dans les systèmes hamiltoniens presque-intégrables.
1.5.2.1. Considérons donc un système hamiltonien presque-intégrable suffisamment
régulier de la forme
H(θ, I) = h(I) + εf(θ, I), (θ, I) ∈ T n × R n ,
où l’on a mis le petit paramètre ε en facteur pour spécifier plus simplement la taille
de la perturbation. Pour m ≥ 1, considérons une action I∗ résonante de multiplicité m,
cela veut dire que sa fréquence ω∗ = ∇h(I∗) est m-résonante. Par un changement de
variables symplectique linéaire, on peut se ramener au cas où ω∗ = (0, ˆω∗) ∈ R m ×R n−m
avec ˆω∗ non résonante, et on va supposer que cette dernière est diophantienne. On peut
alors appliquer la théorie des formes normales résonantes : au voisinage de I∗, en notant
I = (I1, I2) ∈ R m × R n−m et (θ1, θ2) ∈ T m × T n−m , le système est conjugué à
H(θ, I) = h(I) + εg(θ1, I) + µ(ε) ˜ f(θ, I),
avec µ(ε) que l’on peut rendre petit devant ε (exponentiellement petit si le système est
analytique). En négligeant ce dernier terme, on obtient le système moyenné :
[H](I) = h(I) + εg(θ1, I).
Si on note V (θ1) = g(θ1, 0), en effectuant une translation pour ramener I∗ en 0 et un
développement limité en 0, la partie principale du hamiltonien moyenné est décrite par
la somme de deux hamiltoniens
H∗(θ, I) = H1(θ1, I1) + H2(I2) = AI1.I1 + εV (θ1) + ˆω∗I2 + BI2.I2,
où A et B sont des matrices symétriques de taille respective m et n − m (voir [LMS03]
pour plus de détails sur ces transformations). Le hamiltonien H1 décrit un pendule
généralisé (non intégrable si m ≥ 2), et le hamiltonien H2 est un rotateur.
1.5.2.2. Pour les résonances simples (m = 1), on retrouve en particulier l’exemple
d’Arnold (A = 1, V (θ1) = 1 − cos 2πθ1). Plus généralement, si le potentiel V : T −→
R possède un maximum non dégénéré, alors le système H∗ possède dans tout niveau
d’énergie une famille continue de tores de dimension (n−1) partiellement hyperboliques.
Ce sont les systèmes a priori instables, ainsi ces derniers modélisent la dynamique d’un
système presque-intégrable au voisinage d’une résonance simple. Néanmoins, la taille de
ce voisinage n’est que d’ordre √ ε, on ne peut donc pas directement obtenir des résultats
d’instabilité pour les systèmes a priori stables en étudiant les systèmes a priori instables
(voir cependant [Ber09] pour une nouvelle approche).

44 Introduction
Si l’on choisit une perturbation de H∗ qui s’annule le long des tores hyperboliques, on
peut alors appliquer le mécanisme d’Arnold pour montrer l’existence d’orbites de dérive,
mais pour une perturbation "générique", on se retrouve face au "large gap problem".
Cependant, la situation est désormais bien comprise, et tient beaucoup à la remarque
suivante : au lieu de considérer individuellement chaque tore invariant partiellement
hyperbolique, il est plus judicieux de remarquer que leur réunion forme une variété
invariante normalement hyperbolique au sens de [HPS77], et qui va donc se continuer
beaucoup plus facilement aux perturbations du système H∗.
L’utilisation des variétés normalement hyperboliques est un ingrédient essentiel de la
solution au "large gap problem" proposée par Delshams, de la Llave et Seara ([DdlLS06],
voir aussi [DH09] pour un résultat plus général), dans le cas n = 3. En utilisant des outils
de la théorie classique des perturbations (moyennisations résonantes, théorie KAM) ils
arrivent à obtenir une description détaillée de la dynamique restreinte à la variété normalement
hyperbolique perturbée : pour remédier aux problèmes des "trous", créés par
les résonances, ils montrent que ces dernières engendrent d’autres tores invariants (tores
"secondaires") qu’il est possible d’incorporer dans une chaine de transition généralisée.
La dynamique transverse, engendrée par les intersections homoclines et hétéroclines de
la variété normalement hyperbolique, est alors étudiée plus globalement à l’aide de la
"scattering map" (qui n’est en fait pas une application mais une correspondance). Enfin,
la construction d’orbites de dérive s’obtient par un lemme d’inclinaison partiellement hyperbolique,
bien que dans ce cadre il soit plus commode d’utiliser un lemme d’inclinaison
normalement hyperbolique ([ESM10]).
Notons également que d’autres méthodes géométriques ont permis de résoudre le
"large gap problem". D’une part, citons les travaux de Treschev ([Tre04], [PT07]), et
d’autre part, les travaux de Gidea et Robinson ([GR07], [GR09]) où l’on utilise les orbites
d’instabilité de Birkhoff pour traverser les "trous" (dans un cadre très simplifié).
Enfin, suite aux travaux de Mather (en particulier [Mat91] et [Mat93]), signalons que
les meilleurs résultats ont été obtenus par Cheng et Yan ([CY04], [CY09]) et Bernard
([Ber08]).
1.5.2.3. Pour des résonances multiples (m ≥ 2), le système moyenné n’est plus intégrable
et on ne sait dire que bien peu de choses (à ma connaissance), que ce soit sur des
exemples ou pour des classes générales de systèmes hamiltoniens. Pour le moment, on se
limite essentiellement aux résonances doubles dans les systèmes à trois degrés de liberté,
pour lesquels les seuls résultats généraux sont ceux annoncés par Mather ([Mat04]) ainsi
que ceux annoncés très récemment par Marco.
Un cas très particulier et relativement facile de résonances doubles (aussi connu sous
le nom de "systèmes à trois échelle de temps"), qui correspond à l’intersection de deux
résonances simples, l’une "forte" et l’autre "faible", a été étudié dans [Hal95], [Hal97].
Notons aussi un joli exemple d’échange de résonances simples, dû à Bessi ([Bes97a]).
Il construit, toujours par des méthodes variationnelles, une orbite qui dérive le long
d’une résonance simple, s’approche d’une résonance double puis dérive le long d’une
autre résonance simple. Un exemple similaire d’échange de résonances est également
possible par des méthodes géométriques ([Bou09a]).
Enfin, pour conclure, notons bien sûr que c’est la dynamique au voisinage des réso-

1.5 - Exemples d’instabilité 45
nances doubles (et multiples) que l’on doit étudier et comprendre, en vue notamment
de la conjecture de généricité d’Arnold. Il en est de même si l’on cherche à mettre en
évidence des phénomènes d’instabilité plus marquants, comme la question suivante posée
par Herman ([Her98]) : construire un hamiltonien de classe C∞ , Cr-proche pour
r ≥ 2 du hamiltonien intégrable h(I) = 1
2 |I|2 , qui possède une orbite dense sur le niveau
d’énergie H−1 (1) (il s’agit d’une version de l’hypothèse quasi-ergodique). Des progrès
récents dans cette direction ont été effectués d’abord par Kalsohin, Levi et Saprykina
dans [KLS10] où ils construisent une orbite qui est dense dans un sous-ensemble (d’un
niveau d’énergie) de dimension de Hausdorff maximale, puis par Kaloshin, Zhang et
Zheng ([KZZ09]) où ils annoncent la construction d’une orbite qui est dense dans un
sous-ensemble dont la mesure est plus grande que 1
2 .
1.5.3 Temps d’instabilité
On va désormais s’intéresser à des aspects plus quantitatifs sur les phénomènes d’instabilité,
plus précisément à des estimations sur le temps d’instabilité pour des systèmes
analytiques (ou Gevrey).
1.5.3.1. Commençons par le cas a priori stable, c’est-à-dire les perturbations de taille
ε d’un système hamiltonien presque-intégrable.
Rappelons que par le théorème de Nekhoroshev, les solutions sont stables pendant
un temps T(ε) exponentiellement long. On obtient ainsi une minoration sur le temps
d’instabilité
τ(ε) > T(ε) ∼ exp ε −a , a ∼ 1
n 2.
Comme on l’a déjà expliqué, dans le cas quasi-convexe ce résultat fut ensuite amélioré
par Lochak, puis par Marco et Sauzin dans le cas Gevrey. On obtient
τ(ε) > T(ε) ∼ exp ε −a , a = 1
2n ,
et plus généralement dans le cas α-Gevrey, α ≥ 1,
τ(ε) > T(ε) ∼ exp ε −a , a = 1
2αn .
Longtemps avant, après avoir étudié l’exemple d’Arnold, Chirikov ([Chi79], à qui on
doit le terme diffusion d’Arnold) avait prédit que la valeur a = (2n) −1 devait être
optimale en régularité analytique : il a formulé plusieurs conjectures sur les liens entre le
temps d’instabilité et l’écart ("splitting") des variétés invariantes des tores partiellement
hyperboliques, qui suggère un exposant d’instabilité égal à (2n) −1 . On verra dans cette
thèse que ceci n’est pas tout à fait exact.
En ce qui concerne la majoration du temps d’instabilité, on sait aujourd’hui montrer
que
τ(ε) ∼ exp ε −a′
, a ′ 1
=
2(n − 2) .
On doit ce résultat à plusieurs auteurs : Bessi ([Bes96], [Bes97b]) pour les cas n = 3 et
n = 4, Herman, Marco et Sauzin ([MS02]) dans le cas Gevrey non analytique, Lochak et

46 Introduction
Marco ([LM05]) qui ont obtenu (2(n − 3)) −1 dans le cas analytique, et enfin Ke Zhang
([Zha09]) qui a obtenu le résultat général.
En conclusion, l’exposant de stabilité optimal a et l’exposant d’instabilité optimal
a ′ vérifient
1
2n ≤ a < a′ 1
≤
2(n − 2) ,
et dans le cas Gevrey
1
2αn ≤ a < a′ ≤
1
2α(n − 2) .
1.5.3.2. Dans le cas a priori instable, le théorème de Nekhoroshev ne s’applique pas
(directement) et on ne connait donc pas de borne inférieure sur le temps d’instabilité
τ(µ) dans ce contexte. Néanmoins, Lochak a conjecturé que le temps d’instabilité devait
être polynomial en µ (parce que le "splitting" est polynomial en µ), et ceci malgré les
premières estimations super-exponentielles obtenues dans [CG94].
La première estimation réaliste a été obtenue par Marco ([Mar96]), en utilisant la
méthode des fenêtres d’Easton ([Eas78], [Eas81]). En adaptant l’exemple de Bessi, Bernard
a obtenu une majoration de l’ordre de µ −2 ([Ber96], voir également [Cre01] pour
la même estimation avec des méthodes géométriques).
Dans [Loc99], Lochak conjecture que le temps optimal doit être µ −1 ln µ −1 , et ce fut
complètement démontré par Berti, Bolle et Biasco ([BBB03]) par des méthodes variationnelles
introduites par Bessi. Des mécanismes géométriques avec ce temps d’instabilité
ont également été proposés par Cresson et Guillet ([CG03]) ainsi que par Treschev
([Tre04]).
1.6 Au voisinage d’un tore invariant linéairement stable
Pour conclure, mentionnons rapidement un cadre un peu différent, que l’on peut
qualifier de perturbation de système hamiltonien intégrable linéaire : il s’agit d’étudier
la dynamique au voisinage d’un tore invariant quasi-périodique linéairement stable, isotrope
et réductible. On va se contenter des deux cas particuliers extrêmes, à savoir celui
d’un tore invariant quasi-périodique lagrangien (tore de dimension n) et celui d’un point
fixe elliptique (tore de dimension 0), où les hypothèses d’isotropie et de réductibilité sont
automatiques.
1.6.1 Au voisinage d’un tore lagrangien quasi-périodique
1.6.1.1. Commençons par le cas d’un tore invariant quasi-périodique lagrangien. Soit
(M, Ω) une variété symplectique quelconque de dimension 2n, H : M → R un hamiltonien
qui possède un tore invariant lagrangien dont la dynamique est quasi-périodique
de fréquence ω ∈ R n . Il existe donc un plongement Φ : T n → M dont l’image T est
invariante par le champ XH, et qui de plus conjugue la restriction de XH à T à un flot
linéaire de fréquence ω sur T n . Puisque le tore est supposé lagrangien donc isotrope,
par un théorème classique de Weinstein on peut identifier (de manière symplectique) un
voisinage du tore avec T n ×R n et placer le tore en T n ×{0}. Si l’on fait un développement

1.6 - Au voisinage d’un tore invariant linéairement stable 47
limité du hamiltonien dans ces coordonnées (θ, I) ∈ T n × R n , on obtient par invariance
du tore et de la fréquence
H(θ, I) = ω.I + F(θ, I),
où F(θ, I) = O(|I| 2 ). Considérons la partie principale
h(I) = ω.I.
C’est un système intégrable en coordonnées action-angle, dont la fréquence est désormais
constante égale à ω. Le système complet peut se voir comme une perturbation de ce
système intégrable linéaire, la taille de la perturbation étant essentiellement la distance
au tore invariant.
1.6.1.2. Si l’on suppose que ω est non résonante, alors la théorie classique des perturbations
s’applique : pour tout entier m ∈ N ∗ , il existe un difféomorphisme symplectique
Φm de T n × R n qui fixe T n × {0} et tel que
H ◦ Φm(θ, I) = Hm(I) + O(|I| m+1 ),
où Hm est un polynôme de degré m. Ces polynômes Hm, m ∈ N ∗ , sont les invariants de
Birkhoff. En réalité, on peut construire une série formelle H∞(I) ainsi qu’un difféomorphisme
symplectique formel Φ∞ de T n × R n fixant le tore T n × {0} et conjuguant H à
H∞. Cependant, la transformation Φ∞, ainsi que de la forme normale H∞ divergent en
général (ce sont des résultats de Siegel, voir [AKN06], et Pérez-Marco, [PM03]). Il existe
néanmoins deux moyens d’obtenir des informations sur la dynamique au voisinage de ce
tore en utilisant cette forme normale.
1.6.1.3. On peut supposer le système analytique (ou Gevrey) et la fréquence ω diophantienne.
Dans ce cas si ρ est la distance au tore invariant, en choisissant le nombre
de normalisations m = m(ρ) convenablement (tendant vers l’infini lorsque ρ tend vers
0, comme en théorie de Nekhoroshev) on obtient une forme normale avec un reste exponentiellement
petit en ρ. On en déduit facilement un résultat de stabilité pendant un
temps exponentiel : chaque solution qui part à une distance au plus ρ du tore invariant
reste à une distance d’ordre ρ pendant un temps T(ρ) qui est de l’ordre de exp(ρ −1 )
(voir [GG85], [BGG85] et [Fas90]).
1.6.1.4. Une autre possibilité revient à faire un nombre fini m ≥ 2 de normalisations
et à considérer
H ◦ Φm(θ, I) = Hm(I) + O(|I| m+1 ).
On se ramène ainsi à une perturbation d’un système intégrable non linéaire, la partie
intégrable étant donnée par le polynôme Hm. En supposant Hm non dégénéré ou escarpé,
la théorie KAM ou de Nekhoroshev s’applique. La première nous donne l’existence de
beaucoup de tores invariants dans un voisinage, tandis que la seconde donne un résultat
de stabilité pendant un temps exponentiellement long.
1.6.1.5. C’est un fait assez remarquable, constaté par Morbidelli et Giorgilli ([MG95]),
que l’on peut combiner la théorie "linéaire" avec les théories "non linéaires", dans le
sens où l’on peut d’abord construire une forme normale de Birkhoff avec un reste exponentiellement
petit puis ensuite appliquer la théorie KAM ou la théorie de Nekhoroshev
pour obtenir des informations plus précises. On montre alors (assez facilement) que

48 Introduction
dans un petit voisinage du tore invariant, la mesure du complémentaire des tores invariants
est exponentiellement petite (on parle de "condensation exponentielle") et qu’on
a stabilité pendant un temps super-exponentiellement long (on dit que les tores sont
"super-exponentiellement collants").
1.6.2 Au voisinage d’un point fixe elliptique
1.6.2.1. Considérons maintenant le cas d’un point fixe elliptique. Le problème étant
local, on peut se placer sur R 2n muni de sa structure symplectique standard, et placer
le point fixe à l’origine. De plus le hamiltonien étant défini à une constante additive
près, on peut supposer que H(0) = 0, et puisqu’un point fixe pour le flot hamiltonien
n’est rien d’autre qu’un point critique du hamiltonien, on a dH(0) = 0. En faisant un
développement limité du hamiltonien en 0 on obtient
H(z) = H2(z) + O(|z| 3 )
où z est proche de 0 ∈ R 2n , et H2 est la partie quadratique de H en 0. On suppose que
le point fixe est elliptique, autrement dit linéairement (ou plutôt spectralement) stable,
ce qui revient à demander que le spectre de la matrice J0H2 soit purement imaginaire
(J0 étant la structure complexe canonique de R 2n ). Si l’on note {±iα1, . . .,±iαn} ce
dernier, et si l’on suppose qu’il est simple (c’est-à-dire si les αi sont tous distincts), on
peut effectuer un changement de variables linéaire symplectique pour se ramener à
où l’on a noté
H(z) =
Ĩ =
n
i=1
αi
2 (z2 i + z2 n+i ) + O(|z|3 ) = α. Ĩ + O(|z|3 ),
Ĩ(z) = 1
2 (z2 1 + z 2 n+1, . . .,z 2 n + z 2 2n) ∈ R n +
ce que l’on appellera des "actions formelles". Le vecteur α = (α1, . . .,αn) ∈ Rn est appelé
fréquence caractéristique (ou fréquence normale). Considérons le terme principale de H
donnée par
h(z) = h( Ĩ) = α.Ĩ,
alors on peut également le qualifier de système intégrable linéaire, dans le sens où ses
solutions sont toutes quasi-périodiques de même fréquence.
1.6.2.2. Cependant, on remarque que la dimension des tores invariants n’est pas
constante. En effet, le long de chaque solution z(t) les actions formelles sont constantes,
donc si l’on se donne une condition initiale z0 ∈ R 2n et si l’on note Ĩ0 = Ĩ(z0), les
ensembles
T Ĩ 0 = {z ∈ R2n | Ĩ(z) = Ĩ0 }
sont des tores invariants et la dynamique est quasi-périodique. Comme pour les systèmes
intégrables en action-angle, l’espace des phases se décompose en tores invariants
R 2n =
Ĩ 0 ∈R n +
T Ĩ 0,

1.6 - Au voisinage d’un tore invariant linéairement stable 49
mais la dimension des tores n’est plus fixe. Si l’on note Ĩ0 i la i-ème composante de Ĩ0 ,
pour 1 ≤ i ≤ n, on peut écrire
de sorte que
T Ĩ 0 = {z ∈ R2n | z 2 i + z2 n+i = 2Ĩ0 i
dimT Ĩ 0 = card{Ĩ0 i ∈ R+ | Ĩ0 i
, 1 ≤ i ≤ n},
> 0}
varie de 0 (pour le point fixe) à n (tore lagrangien). Les tores de dimension intermédiaire
sont appelés elliptiques ou linéairement stables.
Considérons l’ouvert dense
O = {z ∈ R 2n | Ĩi(z) > 0, 1 ≤ i ≤ n}.
On peut utiliser les coordonnées polaires symplectiques (θ, I) = (θ, Ĩ), où θ ∈ Tn est
défini par
zi = 2Ii cosθi, zi+n = 2Ii sin θi,
pour 1 ≤ i ≤ n. Ce changement de variables symplectique est bien défini et analytique
sur O, et l’on se ramène à un système intégrable en action-angle comme dans la section
précédente (feuilletage en tores lagrangiens). Sur cet ouvert, les actions formelles sont
de "vraies" actions. Cependant, sur le complémentaire de O, le changement de variables
précédent est singulier et cela explique le fait que la dimension des tores varie.
Revenons maintenant à notre système complet
H(z) =
n
i=1
αi
2 (z2 i + z 2 n+i) + O(|z| 3 ) = α. Ĩ + O(|z|3 ),
que l’on voit comme une perturbation du système intégrable, la taille de la perturbation
étant donnée par la distance au point fixe.
1.6.2.3. La théorie de Birkhoff marche de manière identique, bien que l’on ait des
variables cartésiennes z ∈ R 2n et non des variables action-angle (θ, I) ∈ T n × R n ; c’est
en fait dans ce contexte que les formes normales de Birkhoff sont apparues. On dispose
même d’un avantage supplémentaire : pour construire une forme normale de Birkhoff
H ◦ Φm(z) = Hm( Ĩ) + O(|Ĩ|m+1 ),
il suffit de demander une condition de non-résonance à l’ordre 2m sur α, c’est-à-dire
k.α = 0, k ∈ Z n , 0 < |k| ≤ 2m.
Avec une condition de non-résonance à tout ordre, on peut construire les polynômes hm
pour tout m ∈ N ∗ ainsi que la série formelle (généralement divergente) h∞. Avec une
condition diophantienne sur la fréquence, on construit une forme normale avec un reste
exponentiellement petit et on en déduit de la stabilité pendant un temps exponentiellement
long (voir [GDF + 89]).
1.6.2.4. En ce qui concerne la théorie KAM, il est possible d’utiliser les coordonnées
action-angle sur l’ouvert O pour trouver des tores lagrangiens arbitrairement proches du
point fixe elliptique. De plus, les arguments sur la mesure des tores préservés subsistent

50 Introduction
(voir [Pös82]), et on peut également combiner la théorie de Birkhoff et la théorie KAM
pour obtenir des résultats améliorés ([DG96b]).
Pour n = 2, sous la condition de non-dégénérescence iso-énergétique, cela implique la
stabilité du point fixe au sens de Lyapunov ([Arn63b], voir aussi [Arn61] pour d’autres
résultats), mais pour n ≥ 3, le mécanisme d’instabilité d’Arnold fournit des exemples de
points fixes elliptiques topologiquement instables ([DLC83], [Dou88], voir aussi [KMV04]
et [KZZ09]).
Rappelons également que dans le complémentaire de l’ouvert O, le système intégrable
possède des tores invariants linéairement stables dont la dimension n’est pas maximale.
En effet, pour 1 ≤ m < n, si on se donne une condition initiale z0 ∈ R 2n telle que
Ĩ 0 1 , . . .,Ĩ0 m > 0, Ĩ0 m+1 = · · · = Ĩ0 n
alors elle évolue sur le tore TĨ0 de dimension m. On peut alors introduire des variables
action-angle (θ1, . . .,θm, I1, . . .,Im) pour les m premières composantes, et écrire la partie
principale (intégrable) du hamiltonien sous la forme
H(θ, I, z) =
m
i=1
αiIi +
n
i=m+1
= 0,
αi Ĩi, (θ, I, z) ∈ T m × R m × R 2(n−m) .
De manière générale, si un système hamiltonien sur une variété symplectique quelconque
possède un tore invariant quasi-périodique linéairement stable (isotrope, réductible),
alors sa partie principale peut s’écrire sous cette forme. On peut également se demander
si ces tores invariants persistent.
Commençons par le cas particulier m = 1, où l’on a une famille d’orbites périodiques
contenue dans un disque invariant contenant le point fixe elliptique. Si
α1
αi
/∈ Z, i = 2, . . .,n,
alors un théorème de Lyapunov assure que la famille d’orbites périodiques persiste (voir
[SM71]). Dans ce cas particulier, il n’y a pas de petits diviseurs, mais ils apparaissent
pour m > 1.
Pour m = n − 1, le résultat de persistance est dû à Moser ([Mos67]) et dans le
cas plus difficile où m est quelconque, sous des hypothèses diophantiennes adéquats, le
résultat est dû à Eliasson ([Eli88], voir aussi [Pös89]). Ajoutons que tous ces résultats
avaient été annoncés auparavant par Melnikov ([Mel65]).
1.6.2.5. En ce qui concerne la théorie de Nekhoroshev, elle s’applique beaucoup plus
difficilement au voisinage des points elliptiques, en raison de la singularité des variables
action-angle. Ce résultat fut seulement conjecturé par Nekhoroshev, puis partiellement
démontré par Lochak ([Loc92], [Loc95]) dans le cas quasi-convexe. Une preuve complète,
toujours dans le cas quasi-convexe, est due indépendamment à Niederman ([Nie98]) et
Fasso, Guzzo, Benettin ([FGB98], [GFB98]). Les deux preuves utilisent les coordonnées
cartésiennes, la première est une adaptation de la méthode de Lochak dans ce contexte
(voir aussi [Pös99b]) tandis que la seconde utilise les arguments classiques de Nekhoroshev.

2.1 - Résultats de stabilité 51
2 Résultats et questions
Dans cette section, nous allons exposer les résultats de cette thèse de manière assez
informelle (avec un renvoi aux différents chapitres pour des formulations précises), et
nous allons aussi expliquer quelques questions issues de ces travaux.
La plupart des énoncés concernent des systèmes hamiltoniens presque-intégrables de
la forme
H(θ, I) = h(I) + f(θ, I)
|f|D < ε

52 Résultats et questions
Question 1. Donner une condition générique minimale pour avoir stabilité exponentielle.
La condition générique minimale devrait être la suivante : la restriction de h à tout
sous-espace affine, dont la direction est engendrée par des vecteurs à coefficients entiers,
n’admet que des points critiques isolés. C’est une condition nécessaire, d’après les
travaux de Nekhoroshev et Niederman ([Nek79], [Nie06]), il reste donc à montrer que
c’est une condition suffisante, ce qui me semble possible, bien que certains problèmes
"d’uniformité" se posent encore.
Outre le fait d’être plus simple, la preuve proposée dans le troisième chapitre présente
de nombreux avantages, comme de se transposer facilement dans d’autres cadres et de
fournir des résultat plus généraux. Dans le quatrième chapitre, on étudie des résultats de
stabilité au voisinage des points fixes elliptiques et plus généralement des tores invariants
linéairement stables (avec les hypothèse habituelles).
Considérons un point fixe elliptique dans R 2n , avec une fréquence normale α ∈ R n
non résonante à l’ordre 4, et écrivons sa forme normale de Birkhoff :
H(z) = α. Ĩ + βĨ.Ĩ + f(z),
où z ∈ R 2n , β ∈ Sn(R) est une matrice symétrique. On a alors le résultat suivant (voir
Théorème 4.2).
Résultat 3. Pour presque toute matrice symétrique β ∈ Sn(R), le point fixe est exponentiellement
stable.
Notons que ce résultat est plus général que les précédents ([Nie98], [FGB98], [GFB98],
[Pös99b]) qui ne sont valables que si β est définie positive.
De plus, en utilisant une technique de Morbidelli et Giorgilli valable dans le cas
quasi-convexe et pour des tores lagrangiens, on obtient des résultats de stabilité superexponentielle
générique.
On suppose que α est non résonant, et on considère la série formelle de Birkhoff h∞,
on obtient alors le résultat suivant (voir Théorème 4.1).
Résultat 4. Pour un ensemble prévalent de séries formelles h∞, le point fixe est superexponentiellement
stable.
Considérons maintenant un tore invariant lagrangien quasi-périodique, de fréquence
ω ∈ R n diophantienne, et soit h∞ la série formelle de Birkhoff. De manière analogue, on
a le résultat suivant (voir Théorème 4.4).
Résultat 5. Pour un ensemble prévalent de séries formelles h∞, le tore lagrangien est
super-exponentiellement stable.
Enfin on peut également énoncer un théorème généralisant les deux précédents. On
considère un tore invariant linéairement stable (normalement elliptique), isotrope et
réductible. On note ω ∈ R k sa fréquence "interne", α ∈ R l sa fréquence normale et
on suppose que le vecteur (ω, α) ∈ R k+l est diophantien. Dans ce contexte, on peut
également prouver l’existence de formes normales de Birkhoff, et définir la série formelle
h∞. On obtient alors le résultat suivant (voir Théorème 4.5).

2.1 - Résultats de stabilité 53
Résultat 6. Pour un ensemble prévalent de séries formelles h∞, le tore linéairement
stable est super-exponentiellement stable.
Notons que dans les résultats précédents, la condition est formulée dans l’espace
des formes normales (séries formelles) de Birkhoff. On peut alors se poser la question
suivante.
Question 2. Formuler une condition générique dans l’espace des hamiltoniens pour
garantir la stabilité super-exponentielle.
De part l’unicité des invariants de Birkhoff une fois la partie quadratique fixée,
l’application, qui à un germe d’hamiltonien associe la série formelle des invariants de
Birkhoff, est bien définie. Il faudrait maintenant étudier dans quel sens cette application
peut être "régulière".
2.1.2 Cas quasi-convexe
Ici on s’intéresse au cas où la partie intégrable h est quasi-convexe. Si le système
est analytique ou Gevrey, on a des résultats de stabilité exponentielle, mais qui ne sont
pas valables si le système est seulement de classe C k , k ∈ N (la situation devrait être la
même en classe C ∞ , bien que cela ne soit pas démontré).
En revanche, pour des systèmes de régularité finie, on va prouver un résultat de
stabilité polynomiale (Théorème 5.1) dans le cinquième chapitre.
Résultat 7. On suppose que le système est de classe C k , pour k ≥ 3. Si h est quasiconvexe,
et si ε est suffisamment petit, alors le système H = h + f est polynomialement
stable avec les exposants
a =
k − 2 1
, b =
2n 2n .
Autrement dit, il existe des constantes positives c1, c2 qui ne dépendent que h telles
que pour ε suffisamment petit, pour toute action initiale I0,
|I(t) − I0| ≤ c1ε 1
k−2
2n,
−
|t| ≤ c2ε 2n .
Rappelons que dans le cas quasi-convexe, on a des résultats améliorés aux voisinages
des résonances. C’est également vrai en différentiabilité finie (voir Théorème 5.2).
Résultat 8. Pour m ∈ {0, . . .,n−1}, si l’action initiale est proche d’une résonance de
multiplicité m, on peut choisir les exposants
a =
k − 2
, b =
2(n − m)
1
2(n − m) .
Comme en régularité Gevrey, les résultats précédents ne concernent que des hamiltoniens
intégrables quasi-convexes.
Question 3. Démontrer un résultat de stabilité exponentielle (resp. polynomiale) pour
un hamiltonien intégrable générique de classe Gevrey (resp. C k ).
Il n’y aucun doute sur le fait que de tels résultats existent, cependant certaines difficultés
techniques apparaissent lorsque l’on utilise des compositions de moyennisations
(périodiques ou non) en régularité non analytique. Le cas Gevrey semble plus facile.

54 Résultats et questions
2.2 De la stabilité à l’instabilité
On s’intéresse maintenant à la transition entre la stabilité et l’instabilité, c’est-àdire
que l’on cherche un temps de stabilité maximale qui coïnciderait avec le temps
d’instabilité minimale. On se limite au cas où le hamiltonien intégrable h est quasiconvexe.
2.2.1 Cas analytique
On suppose ici que le système est analytique. On a alors stabilité exponentielle avec
des exposants
a = 1 1
, b =
2n 2n .
Dans le sixième chapitre, on va améliorer la valeur de ces exposants (voir Théorème 6.1).
Résultat 9. Pour
0 < δ ≤
on a stabilité exponentielle avec les exposants
a =
1
2n(n − 1) ,
1
− δ, b = δ(n − 1).
2(n − 1)
Pour δ = (2n(n−1)) −1 , on retrouve le résultat précédent et lorsque ce dernier décroît
vers zéro tout en restant positif, l’exposant de stabilité a devient arbitrairement proche
de (2(n − 1)) −1 . En particulier, cela contredit une conjecture de Chirikov ([Chi79]) et
Lochak ([Loc92]), et montre que l’instabilité au sens d’Arnold ne peut pas se produire
loin des résonances.
Rappelons que les exemples d’instabilité donnent a < (2(n − 2)) −1 , un exemple plus
fin devrait certainement donner a < (2(n − 1)) −1 et ainsi notre résultat de stabilité
devrait être optimal.
Question 4. Construire un exemple de solution instable dont le temps est donné par
l’exposant (2(n − 1)) −1 .
2.2.2 Cas Gevrey
On suppose maintenant que le système est de classe α-Gevrey, pour α ≥ 1. On a
alors les exposants de stabilité généralisée
a = 1 1
, b =
2αn 2n ,
que l’on peut également améliorer (Théorème 6.2).
Résultat 10. Pour
0 < δ ≤
1
2αn(n − 1) ,

2.3 - Résultats d’instabilité 55
on a stabilité exponentielle avec les exposants
a =
1
− δ, b =
2α(n − 1)
2δ
5(n − 1) .
Les mêmes remarques s’appliquent en ce qui concerne l’exposant de stabilité a, cependant
on obtient un rayon de confinement (exposant b) moins bon. On peut également
formuler la question suivante, qui devrait être beaucoup plus facile dans le cas non analytique
(α > 1).
Question 5. Construire un exemple de solution instable dont le temps est donné par
l’exposant (2α(n − 1)) −1 .
2.3 Résultats d’instabilité
Enfin, terminons par les résultats d’instabilité. Notre but ici est de construire des
exemples, dans différents contextes, où l’on peut montrer l’existence d’une orbite de
dérive et calculer le temps d’instabilité. Pour simplifier les constructions, on va se limiter
aux perturbations de classe Gevrey non analytiques.
2.3.1 Cas a priori instable
Le premier contexte est celui, bien compris, des systèmes a priori instables. Dans le
septième chapitre, on va construire une orbite qui dérive en un temps optimal (Théorème
7.1).
Résultat 11. Pour α > 1, il existe une perturbation α-Gevrey de taille µ d’un système
a priori instable dont le temps de dérive est
τ(µ) ∼ µ −1 ln µ −1 .
La construction de cet exemple utilise des outils dynamiques, en particulier une
généralisation du théorème de Birkhoff-Smale sur la dynamique chaotique engendrée
par l’intersection transverse des variétés stable et instable d’une variété normalement
hyperbolique.
Avec des méthodes différentes, des estimations similaires ont été données par Berti-
Bolle-Biaso ([BBB03]), Treschev ([Tre04]) ainsi que Cresson et Guillet ([CG03]). De plus,
dans [BBB03], les auteurs démontrent également un résultat de stabilité qui implique
l’optimalité de ce temps d’instabilité en classe analytique. Rappelons que notre exemple
n’est pas analytique, et donc ce résultat de stabilité ne s’applique pas à ce cadre.
Question 6. Montrer que le temps d’instabilité µ −1 ln µ −1 est optimal en régularité
Gevrey pour les perturbations de taille µ des systèmes a priori instables à trois degrés
de liberté.
On peut également poser la question suivante.

56 Résultats et questions
Question 7. Pour k suffisamment grand, montrer que le temps d’instabilité µ −1 ln µ −1
est optimal en régularité C k pour les perturbations de taille µ des systèmes a priori
instables à trois degrés de liberté.
Ces résultats sont sûrement vrai, néanmoins cela nécessite d’établir des estimées de
Nekhoroshev pour des hamiltoniens génériques de classe Gevrey ou C k (encore une fois,
le cas Gevrey est plus facile).
2.3.2 Cas non perturbatif
Le second contexte que l’on va aborder est celui des ε-perturbations de systèmes
hamiltoniens intégrables, avec ε arbitrairement petit seulement si le nombre de degrés
de libertés n est arbitrairement grand. Ce type d’exemple a été introduit par Bourgain
et Kaloshin ([BK05]) pour tester la validité du mécanisme de Nekhoroshev en dimension
infinie : à n fixé, la perturbation n’est pas arbitrairement petite et la théorie de
Nekhoroshev ne s’applique donc pas, mais le caractère perturbatif est restauré lorsque
"n tend vers l’infini" (d’où le terme "high-dimensional Hamiltonian systems" introduit
dans [BK05]). Dans ce papier, les auteurs montrent le résultat suivant : si ε ∼ e −n , alors
il existe une orbite qui dérive linéairement, c’est-à-dire τ(ε) ∼ ε −1 .
Dans le huitième chapitre, en exploitant le mécanisme introduit dans [MS02], on va
démontrer plus simplement une variante du résultat précédent (voir Théorème 8.1).
Résultat 12. Il existe un système hamiltonien presque-intégrable, avec
ε ∼ e −n ln(n lnn) ,
qui possède une orbite dont le temps de dérive est polynomial, plus précisément
τ(ε) ∼ ε −n .
Le temps d’instabilité obtenu est moins bon que dans [BK05], car la taille de la
perturbation est plus petite (notre exemple est "plus perturbatif"). En particulier, si on
note ε0 le seuil d’applicabilité du théorème de Nekhoroshev, on en déduit que
ε0

Part II
Results of stability
Summary
3 Generic exponential stability without small divisors 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.1 Set-up and results . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.2 Comments and prospects . . . . . . . . . . . . . . . . . . . . 65
3.3 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.1 Analytical part . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.2 Geometric part . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.A Proof of the normal form . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.A.1 Preliminary estimates . . . . . . . . . . . . . . . . . . . . . . 82
3.A.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . 84
3.B SDM functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.B.1 Steepness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.B.2 Prevalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4 Generic super-exponential stability for invariant tori 99
4.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Proof of Theorem 4.1 and Theorem 4.2 . . . . . . . . . . . . . . . . . 104
4.2.1 Birkhoff’s estimates . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.2 Nekhoroshev’s estimates and proof of Theorem 4.2 . . . . . . 105
4.2.3 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Further results and comments . . . . . . . . . . . . . . . . . . . . . . 108
4.A Generic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Polynomial stability for C k quasi-convex Hamiltonian systems 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Analytical part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3.1 Elementary estimates . . . . . . . . . . . . . . . . . . . . . . 119
5.3.2 The linear case . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3.3 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
57

3.1 - Introduction 59
3 Generic exponential stability without small divisors
In this chapter, we present a new approach of Nekhoroshev’s theory for a generic unperturbed
Hamiltonian which completely avoids small divisors problems. The proof is an
extension of a method introduced by P. Lochak, it combines averaging along periodic
orbits with simultaneous Diophantine approximation and uses geometric arguments to
handle generic integrable Hamiltonians. This method allows to deal with generic nonanalytic
Hamiltonians and to obtain new results of generic stability around linearly
stable tori. This chapter reproduces the content of [BN09].
3.1 Introduction
3.1.0.1. In this chapter, we are concerned with the stability properties of nearintegrable
analytic Hamiltonian systems. According to a classical theorem of Liouville-
Arnold (see [AKN06]), such systems are locally governed by a Hamiltonian of the form
H(θ, I) = h(I) + f(θ, I)
|f| < ε

60 Generic exponential stability without small divisors
to focus on non-resonant tori to prove that a set of large measure of invariant tori
survives under some regularity and non-degeneracy assumptions. This has now become
a rich and vast subject called KAM theory (see [Pös01], [dlL01] or [Bos86] for some
nice introductions on this theory). Such tori persist in a √ ε-neighbourhood of the
unperturbed ones and therefore for a set of large measure of initial conditions, the
variation of the actions is of order √ ε for all time. But on the other hand, this set of
KAM tori is typically a Cantor family (hence with no interior) and the theory gives
no information on the complement, except when n = 2 where these two-dimensional
invariant tori disconnect the three-dimensional energy level leaving all solutions stable
for all time. However for n ≥ 3, it is still possible to find solutions for which the
variation of the action components is of order one. This was proved by Arnold in
his famous paper ([Arn64]) where he proposed a mechanism to produce examples of
near-integrable Hamiltonian systems where such a drift occurs no matter how small the
perturbation is. This phenomenon is usually referred to Arnold diffusion.
3.1.0.4. Hence for n ≥ 3, results of stability for near-integrable Hamiltonian systems
which are valid for an open set of initial conditions can only be proved over finite times.
This picture was completed by Nekhoroshev in the seventies (see [Nek77],[Nek79] and
[Nie09] for a recent overview of the theory) who proved the following: if the system is
analytic and the unperturbed Hamiltonian h satisfies some quantitative transversality
condition called steepness, then there exist positive constants a, b, ε0, c1, c2 and c3 depending
only on h, such that every solution (θ(t), I(t)) of the perturbed system starting
at time t = 0 satisfies
|I(t) − I(0)| ≤ c1ε b , |t| ≤ c2 exp c3ε −a , (6)
provided that the size of the perturbation ε is smaller than the threshold ε0. The
constants a and b are called the stability exponents. If property (6) is satisfied, we
shall say that the integrable Hamiltonian h is exponentially stable. Hence, KAM and
Nekhoroshev’s theory yield different type of stability results, but they both ultimately
rely on the same tool which is the construction of normal forms, and we shall described
it below.
3.1.0.5. The basic idea is to look at a "more integrable" Hamiltonian which yields a
good approximation of the perturbed system. By the averaging principle (see [AKN06]),
this simpler Hamiltonian is given by the time average of the system along the unperturbed
flow, that is
[H] = h + [f],
where
t
1
[f] = lim f ◦ Φ
t→∞ t 0
h sds
,
and Φ h s is the Hamiltonian flow of the integrable part h. Actually, this average depends
on the dynamics of the unperturbed Hamiltonian and hence on resonant modules associated
to frequencies. So given a submodule M ⊆ Z n , we define its resonant manifold
by
SM = {I ∈ R n | k.∇h(I) = 0 for k ∈ M}.
Due to the ergodic properties of the linear flow with vector ∇h(I) over the torus T n ,
the time average over SM equals the space average along a torus of dimension n − m if

3.1 - Introduction 61
m is the multiplicity of the resonance (i.e. the rank of M), hence n − m angles have
been removed in this case. From a physical point of view, the guiding principle is that
rapidly oscillating terms discarded in averaging cause only small oscillations which are
superimposed to the solutions of the averaged system. In order to prove this claim, one
should check that any solution of the perturbed system remains close to the solution of
the averaged system with the same initial condition. Especially, this will be the case if
one finds a canonical transformation ε-close to identity which conjugates the perturbed
Hamiltonian to its average. Hence we are reduced to a problem of normal form where
one tries to conjugate the system to a simpler one, that is we look for a convenient
system of coordinates.
However, constructing such a good system of coordinates is not an easy task. The
linearised equation of conjugation reads
{χ, h} = f − [f],
if χ is the function generating the conjugation. This is usually called a homological
equation and to solve it we need to invert the linear operator Lh = {., h} acting on a
suitable space of functions. Here our operator is invertible, but its inverse is generally
unbounded: this is the small divisors phenomenon. To see this, just note that once an
action I ∈ SM is fixed (and hence a frequency ω = ∇h(I) satisfying k.ω = 0 for k /∈ M),
the homological equation is a just a first-order, linear with constant coefficients partial
differential equation on T n , namely
ω.∇χ = f − [f].
Such equations are known to be well-suited for Fourier analysis, in our case the operator
Lh is easily diagonalized in a Fourier basis and we find that the eigenvalues are
proportional to the scalar products k.ω, for k ∈ Zn . More precisely, expanding χ and f
as
χ(θ) =
ˆχke i2πk.θ , f(θ) =
ˆfke i2πk.θ ,
then
and so formally
k∈Z n
ˆχk =
[f] =
k∈M
ˆfke i2πk.θ ,
k∈Z n
(i2πk.ω) −1 ˆ fk, k /∈ M,
0, k ∈ M.
The scalar products k.ω appearing in the denominators of (7) are not zero by assumption,
but they can be arbitrarily small and this is inevitable for large integers k. This can
cause the divergence of the Fourier series of χ and hence the unboundedness of the
inverse of Lh. Classical small divisors techniques are concerned with obtaining lower
bounds for these scalar products to ensure the convergence of the series and this leads
necessarily to complicated estimates. Furthermore, to obtain a result applying to all
solutions, a partition of the phase space into resonant manifolds associated to different
modules, usually called the geometry of resonances, has to be achieved and this is a
delicate task. All these techniques are very important, in particular to study Arnold
diffusion and related problems, however we will show that they are not necessary to
prove Nekhoroshev’s estimates.
(7)

62 Generic exponential stability without small divisors
3.1.0.6. Indeed, all these problems are completely bypassed if we only average along
periodic orbits of the unperturbed flow. We first recall the following definition.
Definition 3.1. A vector ω ∈ R n is said to be periodic if there exists a real number
t > 0 such that tω ∈ Z n . In this case, the number
is called the period of ω.
T = inf{t > 0 | tω ∈ Z n }
A basic example is given by a vector with rational components, the period of which is
just the least common multiple of the denominators of its components. Geometrically,
if ω is T-periodic, an invariant torus with a linear flow with vector ω is filled with
T-periodic orbits. In this case, the average along such a periodic solution is given by
1
[f] = lim
t→∞ t
t
0
f ◦ Φ l sds
= 1
T
f ◦ Φ
T 0
l sds, where l denotes the linear Hamiltonian with frequency ω, that is l(I) = ω.I. Then the
homological equation {χ, l} = f − [f] is easily solved without using Fourier expansions
and is given by an explicit integral formula
χ = 1
T
T
0
(f − [f]) ◦ Φ l ssds.
So in this case, there is no small divisors. To understand more concretely the previous
sentence, consider a vector ω ∈ R n and multi-integers k that do not resonate with ω
(that is k /∈ Z n ∩ ω ⊥ ), then in general we don’t have a lower bound on the divisors k.ω
that appears in (7). In that context, small divisors techniques use Diophantine vectors
for which |k.ω| ≥ γ|k| −τ
1 , with γ > 0, τ ≥ 0 and where | . |1 stands for the ℓ 1 -norm,
but nevertheless the lower bound deteriorates as |k|1 increases, causing extra difficulties
(which are usually handled by the so-called ultra-violet cut-off). However if the vector
ω is T-periodic, one simply has |k.ω| ≥ T −1 and the lower bound is uniform in |k|1.
3.1.0.7. Lochak ([Loc92], see also [LN92] and [LNN94] for refinements) has shown that
averaging along the periodic orbits of the integrable Hamiltonian is enough to obtain
Nekhoroshev’s estimates of stability when the unperturbed Hamiltonian is strictly convex
(or strictly quasi-convex, that is its energy sub-levels are strictly convex). Indeed,
using convexity, Lochak obtains open sets around periodic orbits over which exponential
stability holds. Then, Dirichlet’s theorem about simultaneous Diophantine approximation
ensures easily that these open sets recover the whole action space and yields the
global result, avoiding the difficult geometry of resonances. Put it differently, in the
convex case one only needs dynamical informations near resonances of maximal multiplicities,
which are completely characterized by periodic orbits.
The goal of this chapter is to extend Lochak’s approach for a generic set of integrable
Hamiltonians. To do so, we will have to analyze the dynamics in a neighbourhood of
suitable resonances of any multiplicities by using only successive averaging along periodic
orbits together with Dirichlet’s theorem, and this will lead to exponential estimates of
stability for perturbation of a generic integrable Hamiltonian, as stated below.

3.2 - Statement of results 63
Theorem 3.2. Consider an arbitrary real analytic integrable Hamiltonian h defined on a
neighbourhood of a closed ball in R n . For almost any ξ ∈ R n , the integrable Hamiltonian
hξ(x) = h(I) − ξ.I is exponentially stable with the exponents a = b = 3 −1 (2n) −3n .
This will be a direct consequence of Theorems 3.4 and 3.6, see below in section 3.2.1.
This result is not new, see [Nie07b], but the novelty here is our method of proof, which
avoids completely the fundamental problem of small divisors and hence all the associated
technicalities (non-resonant domains, Fourier series, Fourier norm, ultra-violet cut-off
and so on). The analytic part of our proof of Nekhoroshev’s estimates is therefore
reduced to its bare minimum, it is nothing but a classical one-phase averaging, while
our geometric part is based on a clever use of Dirichlet’s theorem along each solution.
Applications of our method to other problems will be discussed below, in section 3.2.2.
To conclude this introduction, we point out that the method of averaging along
periodic orbits has also been used successfully to re-prove recently some KAM theorems
without small divisors (see [KLDM06] and [KLDM07]), even though their techniques
are much more complicated.
3.2 Statement of results
3.2.1 Set-up and results
3.2.1.1. Let B = BR be the open ball centered at the origin of R n of radius R with
respect to the supremum norm, the domain D = T n ×B will be our phase space. To avoid
trivial situations, we assume n ≥ 2. Our Hamiltonian function H is real-analytic and
bounded on D and it admits a holomorphic extension to some complex neighbourhood
of D of the form
Dr,s = {(θ, I) ∈ (C n /Z n ) × C n | |I(θ)| < s, d(I, B) < r},
with two fixed numbers r > 0, s > 0, and where I(θ) is the imaginary part of θ, | . | the
supremum norm on C n and d the associated distance on C n . Equivalently, one can start
with a Hamiltonian H, defined and holomorphic on Dr,s and which preserves reality,
that is H is real-valued for real arguments. Without loss of generality, we may assume
that r < 1 and s < 1. The space of such analytic functions on Dr,s, endowed with the
supremum norm | . |r,s, is obviously a Banach algebra with respect to the multiplication
of functions, and we shall denote it by Ar,s.
Our Hamiltonian H ∈ Ar,s is assumed to be close to integrable, that is of the form
H(θ, I) = h(I) + f(θ, I)
(∗)
|f|r,s < ε 1, that is
where |k|1 = |k1| + · · · + |kn|.
|∂ k h(I)| ≤ M, 1 ≤ |k|1 ≤ 3, I ∈ B,

64 Generic exponential stability without small divisors
3.2.1.2. In order to obtain results of exponential stability, we do need to impose some
non-degeneracy condition on the unperturbed Hamiltonian. Let G(n, k) be the set of all
vector subspaces of R n of dimension k. We endow R n with the Euclidean scalar product,
. stands for the Euclidean norm, and given an integer L ∈ N ∗ , we define G L (n, k) as
the subset of G(n, k) consisting of those subspaces whose orthogonal complement can
be spanned by vectors k ∈ Z n with |k|1 ≤ L.
Definition 3.3. A function h ∈ C 2 (B) is said to be SDM if there exist γ > 0 and
τ ≥ 0 such that for any L ∈ N ∗ , any k ∈ {1, . . ., n} and any Λ ∈ G L (n, k), there exists
(e1, . . .,ek) (resp. (f1, . . .,fn−k)), an orthonormal basis of Λ (resp. of Λ ⊥ ), such that
the function hΛ defined on B by
hΛ(u, v) = h (u1e1 + · · · + ukek + v1f1 + · · · + vn−kfn−k),
satisfies the following: for any (u, v) ∈ B,
for any η ∈ R k \ {0}.
∂uhΛ(u, v) ≤ γL −τ =⇒ ∂uuhΛ(u, v).η > γL −τ η
In other words, for any (u, v) ∈ B, we have the following alternative: either
∂uhΛ(u, v) > γL −τ or ∂uuhΛ(u, v).η > γL −τ η for any η ∈ R k \ {0}. This technical
definition, which is a slight variation of a notion introduced in [Nie07b], is basically a
quantitative transversality condition which is stated in adapted coordinates. It is inspired
on the one hand by the steepness condition introduced by Nekhoroshev ([Nek77])
where one has to look at the projection of the gradient map ∇h onto affine subspaces,
and on the other hand by the quantitative Morse-Sard theory of Yomdin ([Yom83],
[YC04]) where critical or "nearly-critical" points of h have to be quantitatively non
degenerate. The abbreviation SDM stands for "Simultaneous Diophantine Morse" functions,
and we refer to Appendix 3.B for more explanations on this condition and some
justifications on the latter terminology.
3.2.1.3. The set of SDM functions on B with respect to γ > 0 and τ ≥ 0 will be
denoted by SDM τ γ (B), and we will also use the notations
SDM τ (B) =
SDM
γ>0
τ γ (B),
SDM(B) = SDM
τ≥0
τ (B).
The following result states that SDM functions are generic among sufficiently smooth
functions.
Theorem 3.4. Let τ > 2(n 2 + 1) and h ∈ C 2n+2 (B). For Lebesgue almost all ξ ∈ R n ,
the function hξ(I) = h(I) − ξ.I belongs to SDM τ (B).
More precisely, there is a good notion of "full measure" in an infinite dimensional
vector space, which is called prevalence (see [KH08] and [OY05] for nice surveys), and
the previous theorem immediately gives the following result.
Corollary 3.5. For τ > 2(n 2 + 1), SDM τ (B) is prevalent in C 2n+2 (B).
3.2.1.4. Now we can state the main result of this chapter.

3.2 - Statement of results 65
Theorem 3.6. Let H as in (∗) and assume that the integrable part h belongs to
SDM τ γ (B) with τ ≥ 2 and γ ≤ 1. Then there exist positive constants a, b and ε0
depending only on h such that if ε ≤ ε0, for every initial action I(0) ∈ BR/2 the following
estimates
|I(t) − I(0)| < (n 2 + 1)ε b , |t| < exp(ε −a ),
hold true.
More precisely, we can choose the exponents
a = b = 3 −1 (2(n + 1)τ) −n ,
and ε0 depending on the whole set of parameters n, R, r, s, M, γ and τ, but no efforts
was made to improved the stability exponents since the optimality of the constants
involved is not our goal. Actually, this optimality is not relevant for generic integrable
Hamiltonians.
Let us add that the only property used on the integrable part h to derive these
estimates is a specific steepness property, therefore the proof is also valid, and in fact
simpler, assuming the original steepness condition of Nekhoroshev (see Appendix 3.B).
However, note that this is precisely this "weaker" genericity assumption that allows new
results of stability near linearly stable invariant tori (see the next chapter).
We emphasized again that this is not the result itself, but the method of proof which
is new and leads to many improvements as we explain below.
3.2.2 Comments and prospects
To conclude this section we mention other problems for which our method should apply,
mainly the study of elliptic fixed points, Nekhoroshev’s estimates in lower regularity
and finally estimates in large or infinite dimensional Hamiltonian systems. In all these
topics, the method of periodic averaging have already proved to be very useful.
3.2.2.1. First our analytic arguments are very intrinsic and this is important in the
study of the stability of elliptic fixed points in Hamiltonian systems. Actually, in this
case the transformation in action-angle variables (via the symplectic polar coordinates)
admits singularities which do not allow to derive directly stability results from Nekhoroshev’s
theory. In the convex case, this problem has been overcomed independently by
Fassò, Guzzo and Benettin([FGB98], [GFB98]) and by Niederman ([Nie98]). Both use
Cartesian coordinates, the first study uses the classical approach and adapted Fourier expansions
while the second one relies on periodic averaging and simultaneous Diophantine
approximation. The latter proof was clarified by Pöschel ([Pös99b]). With our approach,
we can remove the convexity hypothesis to have exponential stability around an elliptic
fixed point under a generic assumption on the non-linear part. Furthermore, assuming
a Diophantine condition on the normal frequency it is well-known since Morbidelli and
Giorgilli ([MG95]) that one can even obtain super-exponential stability by combining a
sufficiently large number of Birkhoff normalizations with Nekhoroshev’s estimates. Here,
with our method generic results of super-exponential stability around elliptic fixed points
are also available, and similarly around invariant Diophantine Lagrangian tori and even
isotropic reducible linearly stable tori. All this results are contained in the next chapter.

66 Generic exponential stability without small divisors
3.2.2.2. Furthermore, one should mention that periodic averaging are well-suited for
non-analytic Hamiltonians and our formalism should also carry on in this context. The
advantage of periodic averaging is clear already at the linear level when solving the
homological equation: if the system is of finite differentiability, then the solution of the
homological equation when expanded in Fourier series is subjected to a disastrous loss
of derivatives (larger than the number of degrees of freedom), while with the explicit
integral formula, this loss of derivatives is minimal. Hence we can expect stability estimates
in finite differentiability, for both convex and generic unperturbed Hamiltonian,
but of course with polynomial bound on the time of stability. Note that the analyticity
of the studied system is only needed for the construction of normal forms up to an exponentially
small remainder, but our steepness condition is generic for Hamiltonians of
finite but sufficiently high regularity. Concerning Gevrey regularity, Marco and Sauzin
([MS02]) have already proved exponential estimates of stability in the convex case and
for the C k regularity, polynomial estimates of stability will be proved in the fifth chapter.
With our method these results should also hold for a generic integrable Hamiltonian. It
can also be noticed that the analytical properties of the expansions arising in periodic
averaging are accurately known ([Nei84],[RS96])
3.2.2.3. Finally, results of stability for large Hamiltonian systems as a model for statistical
mechanics have been obtained by Bambusi and Giorgilli ([BG93]) and Bourgain
([Bou04]), and for non-linear evolution PDE seen as an infinite dimensional Hamiltonian
system mostly by Bambusi ([Bam99], [BN02]) and then clarified by Pöschel ([Pös99a]).
All these works use Lochak’s approach in the convex case. We believe that our method
should allow to remove the convexity assumption in those results to obtain more general
statements. Actually, our approach rely on a property of composition of averaging
transformations which was already used by Bambusi ([Bam99]).
3.2.2.4. This chapter is organized as follows. In the next section, we state our normal
form and explain the main ideas, and then we give the proof of Theorem 3.6. The
complete proof of the normal form is deferred to Appendix 3.A, and in Appendix 3.B
we collect the basic properties of SDM functions that we shall need and we prove Theorem
3.4 and Corollary 3.5.
3.2.2.5. In the text, we shall adopt the following notation taken from [Pös99b]: we
will write u

3.3 - Proof of Theorem 3.6 67
with H ∈ Ar,s. As usual, the proof of exponential stability estimates splits into an
analytic part and a geometric part.
The analytic part is contained in section 3.3.1. It consists in the construction of normal
forms on a neighbourhood of specific resonances, that is suitable coordinates which
display the relevant part of the perturbation on such a neighbourhood. Basically, we will
reduce the perturbation to a so-called resonant term which is dynamically significant,
and a general term which will only cause exponentially small deviations.
The geometric part is expanded in section 3.3.2, and it is mainly based on the
properties of the underlying integrable system. The strategy will be first to defined a
class of solutions, which we call restrained, and for which it is obvious from our normal
forms that they are stable for an exponentially long time. Using this intermediate result,
we will then show that all solutions are in fact exponentially stable, and our main tools
to do this will be an adapted steepness property satisfied by our integrable system, as
well as a basic theorem of Dirichlet on simultaneous Diophantine approximation.
3.3.1 Analytical part
3.3.1.1. Let us begin by describing the neighbourhoods of resonances we will consider.
Given a sequence of linearly independent periodic vectors (ω1, . . ., ωn), with periods
(T1, . . .,Tn), we define in the complex phase space, for j ∈ {1, . . .,n}, the domains
Drj,sj (ωj) = {(θ, I) ∈ Drj,sj | |∇h(I) − ωj|

68 Generic exponential stability without small divisors
We will write lj for the linear integrable Hamiltonian with frequency ωj, that is
lj(I) = ωj.I for j ∈ {1, . . .,n}. For any function f, we will denote [f]j its average along
the periodic flow generated by lj, that is
[f]j = 1
Tj
Tj
0
f ◦ Φ lj
s ds.
3.3.1.3. Our interest here is to obtain normal forms on nearly-periodic tori up to an
exponentially small remainder with respect to some parameter m ∈ N, that we will
choose later of order ε −1 (during the proof of Theorem 3.6). To this end, we will need
the following conditions (Aj), for j ∈ {1, . . .,n}, where (A1) is
and for j ∈ {2, . . ., n}, (Aj) is
mT1ε ·

3.3 - Proof of Theorem 3.6 69
with {gj, li} = 0 for i ∈ {1, . . ., j} and the estimates
Moreover, we have Ψj = Φ1 ◦ · · · ◦ Φj with
|∂θgj|2rj/3,2sj/3

70 Generic exponential stability without small divisors
Remark 3.8. Note that this property was actually used by Bambusi ([Bam99], Lemma
8.4).
3.3.1.5. Let us now examine the dynamical consequences of our normal form. As usual,
it will be used to control the directions, if any, in which the action variables in these
new coordinates can actually drift, and we shall come back to our original coordinates
at the beginning of section 3.3.2.
Under the assumptions of Proposition 3.1, consider the Hamiltonian
Hj = H ◦ Ψj = h + gj + fj
on the domain D2rj/3,2sj/3(ωj). Let Mj be the Z-module
Mj = {k ∈ Z n | k.ωi = 0, i ∈ {1, . . ., j}},
whose rank is n − j, and Λj = Mj ⊗ R the vector space spanned by Mj.
The following lemma is completely obvious using the definition of the Poisson
bracket.
Lemma 3.9. The equality {gj, li} = 0, for all i ∈ {1, . . ., j}, is equivalent to ∂θgj ∈ Λj.
Now consider a solution (θ j (t), I j (t)) of Hj with an initial action I j (tj) ∈ B2rj/3(ωj)
for some tj ∈ R, and define the time of escape of this solution as the smallest time
˜tj ∈]tj, +∞] for which I j (˜tj) /∈ B2rj/3(ωj). The only information we shall use from our
normal form is contained in the next proposition.
Proposition 3.2. Let Πj be the projection onto the linear subspace Λj, then with the
previous notations, we have
In particular,
|I j (t) − I j (tj) − Πj(I j (t) − I j (tj))|

3.3 - Proof of Theorem 3.6 71
and therefore
for t ∈ [tj, e m [∩[tj, ˜tj[.
|I j (t) − I j (tj) − Πj(I j (t) − I j (tj))|

72 Generic exponential stability without small divisors
Given a solution (θ(t), I(t)) ∈ B starting at time t0 = 0, we can define inductively
the "averaged" solution (θ i (t), I i (t)) for i ∈ {1, . . .,n} by
Φi(θ i (t), I i (t)) = (θ i−1 (t), I i−1 (t))
as long as I i−1 (t) ∈ Bi, with (θ 0 (t), I 0 (t)) = (θ(t), I(t)). Moreover, using our estimate
on Φi we have
|I i (t) − I i−1 (t)| · 0 and m ∈ N, a solution (θ(t), I(t)) of the Hamiltonian
(∗), starting at time t0 = 0, is said to be restrained (by r0, up to time e m ) if we
can find sequences of:
(1) radii (r1, . . .,rn), with 0 < rn < · · · < r1 < r0;
(2) widths (s1, . . .,sn), with 0 < sn < · · · < s1;
(3) independent periodic vectors (ω1, . . .,ωn), with periods (T1, . . .,Tn);
(4) times (t1, . . .,tn), with 0 = t0 ≤ t1 ≤ · · · ≤ tn ≤ tn+1 = e m ,
satisfying, for j ∈ {0, . . ., n − 1}, conditions (Aj+1) and the following conditions (Bj)
defined by
|Ij (t) − Ij (tj)| < rj, t ∈ [tj, tj+1],
|∇h(Ij (tj+1)) − ωj+1| < rj+1.
(Bj)
Before explaining this definition, we need to make several remarks. First, for j ∈
{0, . . .,n−2} we will see that the first condition of (Bj+1) is well defined by the second
condition of (Bj). Furthermore, for j ∈ {0, . . .,n−1} the last condition in (Bj) implies
in particular that the set Bj+1(ωj+1) is non-empty so we may remove this assumption
from (Aj+1). Finally, we can choose the same sequence of widths (s1, . . .,sn) for all
solutions, therefore we may already fix si ·=s with a suitable positive constant and this
simplifies some conditions (for instance, the condition mTjrj ·< sj appearing in (Aj) will
be replaced by mTjrj ·< 1).
We have chosen the word "restrained" because for such a solution the actions I(t)
(or some properly normalized actions I j (t)) are forced to pass close to a resonance at the
time t = tj, the multiplicity of which decreases as j increases, and moreover the variation
of these (normalized) actions is controlled on each time interval [tj, tj+1]. Hence after
the time tn, the actions are in a domain free of resonances and they are easily confined
in view of the last part of Proposition 3.2. This is reminiscent of the original mechanism
of Nekhoroshev, but the fact that we consider each solution individually will greatly
simplify this geometric part.
3.3.2.3. Let us see how the actions of a restrained solution are easily confined for an
exponentially long time with respect to m. We shall write
for j ∈ {1, . . ., n}.
ρj = r1 + · · · + rj,

3.3 - Proof of Theorem 3.6 73
Proposition 3.3. Consider a restrained solution (θ(t), I(t)), with an initial action
I(0) ∈ BR/2. If
(i) ε ·

74 Generic exponential stability without small divisors
3.3.2.4. Restrained solutions are exponentially stable, and now we will show that this
is in fact true for all solutions. However, to use our steepness arguments this will be
done quite indirectly, and so it is useful to introduce the following definition.
Definition 3.11. Given r0 > 0 and m ∈ N, a solution (θ(t), I(t)) of the Hamiltonian
(∗), starting at time t0 = 0, is said to be drifting (to r0, before time e m ) if there
exists a time t∗ satisfying
|I(t∗) − I(0)| = (n 2 + 1)r0, 0 < t∗ < e m .
Of course, this definition makes sense only if (n 2 + 1)r0 < R/2. In view of Proposition
3.3, drifting solutions cannot be restrained. However, we will prove below that if
such a drifting solution exists, it has to be restrained under some assumptions on r0, m
and ε, which will eventually prove that all solutions are in fact exponentially stable.
More precisely, assuming the existence of a drifting solution, we will construct a
sequence of radii (r1, . . .,rn), an increasing sequence of times (t1, . . .,tn) and a sequence
of linearly independent vectors (ω1, . . ., ωn), with periods (T1, . . .,Tn) satisfying, for
j ∈ {0, . . ., n−1}, assumptions (Aj+1) and (Bj). All sequences will be built inductively,
and we first describe the tools that we shall need.
3.3.2.5. For j ∈ {1, . . .,n}, recall that Λj is the vector space spanned by
Mj = {k ∈ Z n | k.ωi = 0, i ∈ {1, . . ., j}},
and that Πj (resp. Π ⊥ j ) is the projection onto Λj (resp. Λ ⊥ j ). Let us define the integer
Lj = sup {|Tiωi|} ∈ N
i∈{1,...,j}
∗ , j ∈ {1, . . ., n − 1}.
For completeness, we set Λ0 = Rn , L0 = 1 and in this case Π0 is nothing but the
identity. To construct the sequence of times, we will rely on the fact that our integrable
part h belongs to SDM τ γ (B), so that it satisfies the following steepness property (see
Appendix 3.B).
Lemma 3.12. For j ∈ {0, . . .,n − 1}, let λj be any affine subspace with direction Λj,
and take r < 1. Then for any continuous curve Γ from [0, 1] to Λj ∩ B with length
|Γ(0) − Γ(1)| = r ·r 2 .
Proof. For any j ∈ {1, . . ., n − 1}, the orthogonal complement of Λj is spanned by
ω1, . . .,ωj, hence by the integer vectors T1ω1, . . .,Tjωj, so that Λj belongs to G Lj (n, n−j)
with the integer Lj defined above. Therefore one can apply the Proposition 3.7 in
Appendix 3.B to get the required properties (note that here we are using the supremum
norm instead of the Euclidean norm, so the implicit constants are different).
For j = 0, the curve Γ goes from [0, 1] to B = B ∩ R n , but since the orthogonal
complement of R n is trivial one can take L0 = 1.

3.3 - Proof of Theorem 3.6 75
3.3.2.6. To construct the sequence of periodic vectors, we shall use the following
lemma, which is a straightforward application of Dirichlet’s theorem on simultaneous
Diophantine approximation (see [Cas57]).
Lemma 3.13. Given any vector v ∈ R n and any real number Q > 0, there exists a
T-periodic vector ω satisfying
|v − ω| ≤ T −1 1
−
Q n−1, |v| −1 ≤ T ≤ Q|v| −1 .
Proof. Fix any real number Q > 0. We can write the vector v, up to re-ordering its
components, as v = |v|(±1, x) with x ∈ R n−1 , and it will be enough to approximate x by
a periodic vector. By a theorem of Dirichlet, we can find an integer q, with 1 ≤ q < Q,
such that
1
−
|qx − p| ≤ Q n−1,
for some p ∈ Z n . The vector q −1 p is trivially q-periodic, hence the vector ω =
|v|(±1, q −1 p) is T-periodic, with T = |v| −1 q, therefore
and we have the estimate
|v| −1 ≤ T ≤ Q|v| −1 ,
|v − ω| ≤ T −1 |qx − p| ≤ T −1 1
−
Q n−1.
3.3.2.7. Now we can finally prove that drifting solutions are in fact restrained under
some assumptions. This will be done inductively, and for technical reasons we separate
the first step (Proposition 3.4) from the general inductive step (Proposition 3.5).
Proposition 3.4. Let (θ(t), I(t)) be a drifting solution. If
(i) r0 ·

76 Generic exponential stability without small divisors
Now using the fact that h ∈ SDM τ γ (B) and r0 ·< γ (recall that L0 = 1), we can apply
Lemma 3.12 (the case j = 0) to the curve Γ1 restricted to [0, t ∗ 0] to find a time t1 ∈ [0, t ∗ 0]
for which
|I(t) − I(0)| < r0, t ∈ [0, t1],
|∇h(I(t1))| ·>r 2 0.
The first inequality of (12) gives (a).
Now choose Q1 = ε −a1(n−1) , for some positive constant a1 yet to be chosen, and apply
Lemma 3.13 to approximate ∇h(I(t1)) by a T1-periodic vector ω1, that is
Moreover, since
|∇h(I(t1)) − ω1| ≤ T −1
1 Q
1
− n−1
1
r 2 0

3.3 - Proof of Theorem 3.6 77
Proof. First note that for j = 1, we do not require that t1, ω1 and r1 satisfy (A1) since
this is implied by the conditions (i), (ii) and (iii), and for j > 1, the same conditions
reduce assumption (Aj+1) to the inclusion of real domains Brj+1 (ωj+1) ⊆ B2rj/3(ωj)
(recall that by condition (Bj−1) these domains are non-empty, and that we have already
fixed sj ·= s).
Therefore, we need to construct tj+1, ωj+1 and rj+1 satisfying
(a) |I j (t) − I j (tj)| < rj, t ∈ [tj, tj+1];
(b) |∇h(I j (tj+1)) − ωj+1| < rj+1;
(c) ωj+1 is independent of (ω1, . . .,ωj);
(d) Brj+1 (ωj+1) ⊆ B2rj/3(ωj),
and the estimates (14).
Let ˜tj be the maximal time of existence within Bj of the solution I j (t) starting at
I j (tj). Since (Aj) is satisfied, we can apply Proposition 3.2 and for t ∈ [tj, ˜tj] ∩ [tj, e m ],
we have
|I j (t) − I j (tj) − Πj(I j (t) − I j (tj))| ·rj. (16)

78 Generic exponential stability without small divisors
But conditions (v) and (vi) give in particular
so that (15) and (16) yields
ε ··rj.
Now using (v) again, this gives
|Πj(I j (˜tj) − I j (tj))| >· TjrjL −1
τ j ,
and so we can certainly find a time t ∗ j ∈ [tj, ˜tj] such that
|Πj(I j (t ∗ j ) − Ij (tj))| = TjrjL −1
τ j .
Second case: ˜tj > e m . We will first prove that t∗ ∈ [tj, e m ]. Indeed, otherwise t∗
belongs to [tk, tk+1] for some k ∈ {0, . . ., j − 1} and we can write
k−1
|I(t∗) − I(0)| ≤ |I(t∗) − I(tk)| + |I(ti+1) − I(ti)|. (17)
Each term of the right-hand side of (17) is easily estimated: using (Bi) for i ∈ {0, . . ., k−
1} we have
|I i (ti+1) − I i (ti)| < ri, |I k (t∗) − I k (tk)| < rk,
which implies, by the triangle inequality and the estimate (8)
Moreover,
hence we find
|I(ti+1) − I(ti)| < 2ρi + ri, |I(t∗) − I(tk)| < 2ρk + rk.
|I(t∗) − I(0)| <
i=0
|I(t1) − I(0)| < r0,
k
i=1
(2ρi + ri) + r0 < (n 2 + 1)r0,
which of course contradicts the definition of our drifting time t∗.
Now to prove the claim, we argue by contradiction and suppose that
|Πj(I j (t) − I j (tj))| < TjrjL −1
τ j , t ∈ [tj, e m ].
Since t∗ ∈ [tj, e m ], we can use the previous inequality together with the estimate (15)
and both conditions (v) and (vi) to first obtain
and then with the triangle inequality
|I j (t∗) − I j (tj)| < rj,
|I(t∗) − I(tj)| < 2ρj + rj.

3.3 - Proof of Theorem 3.6 79
Now, as the argument above, writing
j−1
|I(t∗) − I(0)| ≤ |I(t∗) − I(tj)| + |I(ti+1) − I(ti)|
we find the same contradiction on the time t∗, which completes the proof of the claim.
Now consider the restriction of the curve Γj+1 on the interval [tj, t∗ j]. Using our
claim together with conditions (iv) and (v), we can apply Lemma 3.12 to find a time
tj+1 ∈ [tj, t∗ j ] such that
|Πj(I j (t) − Ij (tj))| < TjrjL −1τ
j , t ∈ [tj, tj+1],
|Πj(∇h(Γj+1(tj+1)))| ·> TjrjL −1
2τ (18)
j .
The first inequality of (18), together with (15) and conditions (v) and (vi) give
i=0
|I j (t) − I j (tj)| < rj
for t ∈ [tj, tj+1], hence (a) is verified. Now as in the first step, choose Qj+1 = ε −aj+1(n−1)
for some positive constant aj+1 to be chosen later, and apply Lemma 3.13 to approximate
∇h(I j (tj+1)) by a Tj+1-periodic vector ωj+1, that is
|∇h(I j (tj+1)) − ωj+1| ≤ T −1
1
n−1
j+1Q− j+1
= T −1
j+1 εaj+1 . (19)
Let rj+1 =·T −1
j+1 εaj+1 so that (b) is verified by (19). To estimate the period Tj+1 and the
number Lj, we need a lower bound for |∇h(I j (tj+1))| and we will use the fact that we
have such a lower bound for |∇h(I(t1)| (see the second inequality of (12)). First note
that one has easily
|I j (tj+1) − I(t1)|

80 Generic exponential stability without small divisors
The estimates (21) and (23) gives (14) (note that we have a similar estimate for Lj+1,
however at the end we shall only need estimates for L1, . . .,Ln−1).
Next having built rj+1, we need to check that ωj+1 is independent of (ω1, . . .,ωj).
First, by using the mean value theorem, the estimate (15) and our condition (vi), we
have
|∇h(I j (tj+1)) − ∇h(Γj+1(tj+1))| ·< TjrjL −12τ
j ,
and together with the second estimate of (18), this gives
Furthermore, using (19)
hence with (viii), we get
|Πj(∇h(I j (tj+1)))| ·> TjrjL −12τ
j . (24)
|Πj(∇h(I j (tj+1)) − ωj+1)| ≤ |∇h(I j (tj+1)) − ωj+1| · TjrjL −12τ
j
and so Πj(ωj+1) is non zero, which means that ωj+1 is not a linear combination of
{ω1, . . .,ωj}. This proves (c).
Finally we can write
and hence
|ωj+1 − ωj| ≤ |ωj+1 − ∇h(I j (tj+1))| + |∇h(I j (tj+1)) − ∇h(I j (tj))|
So given any I ∈ Brj+1 (ωj+1), we have
+ |∇h(I j (tj)) − ∇h(I j−1 (tj))| + |∇h(I j−1 (tj)) − ωj|,
|ωj+1 − ωj|

3.3 - Proof of Theorem 3.6 81
(iii) TjrjL −1τ
j ·< rj, for j ∈ {1, . . .,n − 1};
(iv) mTjrj ·

82 Generic exponential stability without small divisors
So we need to choose positive constants aj, j ∈ {1, . . ., n}, a and b such that the previous
conditions are satisfied. First note that by (i ′ ), the sequence aj, for j ∈ {1, . . ., n}, has
to be increasing, hence
max ai = an.
i∈{1,...,j}
Then using (v ′ ), we observe that (i ′ ) is satisfied if aj+1 = 2τ(n+1)aj for j ∈ {1, . . ., n−1},
that is
aj = (2τ(n + 1)) j−1 a1.
Now for (ii ′ ) to be satisfied, one can choose
so aj, for j ∈ {2, . . ., n}, is determined by
Then, since τ ≥ 2, we may choose
a1 = (2τ(n + 1)) −n ,
aj = (2τ(n + 1)) −n−1+j .
b = 3 −1 a1 = 3 −1 (2τ(n + 1)) −n
and (iii ′ ) easily holds. Finally, we may also choose
a = b = 3 −1 (2τ(n + 1)) −n
so that (iv ′ ) is satisfied. With those values, it is easy to check that (v ′ ), (vi ′ ) and (vii ′ )
holds, recalling that τ ≥ 2 and n ≥ 2. To conclude, just note that (viii ′ ), (ix ′ ), (x ′ ) and
(xi ′ ) are satisfied if ε ≤ ε0 with a sufficiently small ε0 depending on n, R, r, s, M, γ and
τ. This ends the proof.
3.A Proof of the normal form
In this first appendix we will give the proof of the normal form 3.1. We will closely
follow the method of [Pös99b] and deduce our result from an equivalent version in terms
of vector fields (Proposition 3.6 below).
3.A.1 Preliminary estimates
Before giving the proof, we will need some general estimates based on the classical
Cauchy inequality.
3.A.1.1. First consider the case of a function f analytic on some domain Dr,s, and
recall that
|∂θf|r,s = max
1≤i≤n |∂θif|r,s, |∂If|r,s = max |∂Iif|r,s. 1≤i≤n
We take r ′ , s ′ such that 0 < r ′ < r and 0 < s ′ < s. The first estimate is classical, but
we repeat the proof for convenience.
Lemma 3.14. Under the previous assumptions, we have
|∂If|r−r ′ ,s < 1
r ′ |f|r,s,
1
|∂θf|r,s−s ′ <
s
′ |f|r,s.

3.A - Proof of the normal form 83
Proof. For x = (θ, I) ∈ Dr−r ′ ,s and any unit vector v ∈ C n , consider the function
Fx,v : t ∈ C ↦−→ f(θ, I + tv) ∈ C.
This function is well-defined and holomorphic on the disc |t| < r ′ , so the classical Cauchy
estimate gives
|F ′ x,v(0)| < 1
|f|r,s,
r ′
from which the inequality for ∂If follows easily by optimizing with respect to x and v.
The estimate for ∂θf is completely similar.
3.A.1.2. Now let j ∈ {1, . . .,n}, and let f and g be analytic functions defined on the
domain
Drj,sj (ωj) = {(θ, I) ∈ Drj,sj | |∇h(I) − ωj| 1, so we have the inequality
|Xf|rj,sj ≤ ||Xf||rj,sj and the equality holds if f is integrable. Moreover, note that
−1
each norm || . ||rj,sj is normalized with s1r1 (and not with sjr −1
j ): by our inclusions of
domains, this implies in particular that || . ||rj+1,sj+1 ≤ || . ||2rj/3,2sj/3.
It is well-known how to use the Cauchy inequality to estimate the size of the Poisson
bracket {f, g} in terms of f and g. Similarly, our second estimate is concerned with
the size of the vector field [Xf, Xg] in terms of Xf and Xg. We take r ′ , s ′ such that
0 < r ′ < rj and 0 < s ′ < sj.
Lemma 3.16. Under the previous assumptions, we have
and moreover, if g is integrable, then
Proof. First recall that
||[Xf, Xg]||rj−r ′ 1
,sj−s ′ < ||Xf||rj,sj ||Xg||rj,sj ,
r ′
||[Xf, Xg]||rj−r ′ 1
,sj−s ′ < ||Xf||rj,sj ||Xg||rj .
s ′
[Xf, Xg] = d
dt (Φg
t) ∗ Xf
t=0
Now fix x ∈ Drj−r ′ ,sj−s ′, and let us define the vector-valued function
Fx : t ∈ C ↦−→ (Φ g
t) ∗ Xf(x) ∈ C 2n .
.

84 Generic exponential stability without small divisors
Clearly, the map Φ g
t is analytic, and it sends Drj−r ′ ,sj−s ′(ωj) into Drj,sj (ωj) for complex
values of t satisfying
|t| < r ′ ||Xg|| −1
rj,sj ,
hence the function Fx is well-defined and analytic on the disc |t| < r ′ ||Xg|| −1 . So rj,sj
applying the classical Cauchy estimate to each component of Fx and optimizing with
respect to x ∈ Drj−r ′ ,sj−s ′ we obtain the desired inequality
||[Xf, Xg]||rj−r ′ 1
,sj−s ′ < ||Xg||rj,sj ||Xf||rj,sj .
r ′
In case g is integrable, the map Φ g
t leaves invariant the action components, so the same
reasoning can be applied on the larger disc
giving the improved estimate
3.A.2 Proof of Proposition 3.1
|t| < s ′ ||Xg|| −1
rj,sj ,
||[Xf, Xg]||rj−r ′ 1
,sj−s ′ < ||Xf||rj,sj ||Xg||rj .
s ′
Now we can pass to the proof of Proposition 3.1. Given ˜ε > 0 which will be the size of
our perturbating vector field Xf, let us introduce a slightly modified set of conditions
( Ãj), for j ∈ {1, . . .,n}, where ( Ã1) is
mT1˜ε ·< r1, mT1r1 ·< s1, 0 < r1

3.A - Proof of the normal form 85
with {gj, li} = 0 for i ∈ {1, . . ., j}, and the estimates
||Xgj ||2rj/3,2sj/3

86 Generic exponential stability without small divisors
Lemma 3.17 (First iterative lemma). Consider H = h+g+f on the domain Dr1,s1(ω1),
with h integrable, {g, l1} = 0, and assume that
If we have
||Xg||r1,s1

3.A - Proof of the normal form 87
and hence ||Xχ||r1,s1 < T1˜ε. By the hypothesis T1˜ε < r ′ < s ′ our transformation ϕ1
maps Dr1−r ′ ,s1−s ′(ω1) into Dr1,s1(ω1) and
|ϕ1 − Id|r1−r ′ ,s1−s ′ < T1˜ε.
Therefore it remains to estimate the vector field
1
Xf+ = (Φ
0
χ
t ) ∗ [Xh−l1 + Xg + Xft, Xχ]dt,
and for that it is enough to estimate the brackets [Xft, Xχ], [Xg, Xχ] and [Xh−l1, Xχ].
Using Lemma (3.16), we find
and
||[Xft, Xχ]||r1−r ′ 1
,s1−s ′ <
r ′ ||[Xft||r1,s1||Xχ||r1,s1

88 Generic exponential stability without small divisors
Proof. Our Hamiltonian is H = h + g + f, h is integrable and we have {g, li} = 0 for
i ∈ {1, . . .,j + 1} and {f, li ′} = 0 for i′ ∈ {1, . . ., j}. Once again, our transformation
ϕj+1 = Φ χ
1 will be the time-one map of the Hamiltonian flow generated by some auxiliary
function χ.
We choose
χ = 1
Tj+1
Tj+1
0
(f − [f]j+1) ◦ Φ lj+1
t tdt, (28)
where [.]j+1 is the averaging along the Hamiltonian flow of lj+1. Introducing the notation
ft = tf + (1 − t)[f]j+1, like in Lemma 3.17 we have
with
g+ = g + [f]j+1, f+ =
H ◦ ϕj+1 = h + g+ + f+
1
0
{h − lj+1 + g + ft, χ} ◦ Φ χ
t dt.
We need to verify that we still have {g+, li} = 0 for i ∈ {1, . . .,j + 1} and {f+, li ′} = 0
for i ′ ∈ {1, . . ., j}. By definition, {[f]j+1, lj+1} = 0, and for i ′ ∈ {1, . . ., j}, we compute
{[f]j+1, li ′} =
=
=
= 0.
1
Tj+1
{f ◦ Φ lj+1
t , li ′}dt
Tj+1 0
Tj+1 1
{f ◦ Φ
Tj+1 0
lj+1
t , li ′ ◦ Φlj+1 t }dt
Tj+1
1
Tj+1
0
{f, li ′} ◦ Φlj+1 t dt
This proves that {g+, li} = {g + [f]j+1, li} = 0 for i ∈ {1, . . .,j + 1}. Now a completely
similar calculation shows that for i ′ ∈ {1, . . ., j}, {χ, li ′} = 0, hence li ′ ◦ Φχt
= li ′ and
therefore
{f+, li ′} =
1
0
{{h − lj+1 + g + ft, χ}, li ′} ◦ Φχt
dt.
The double bracket in the expression above is zero, as a consequence of Jacobi identity
and the fact that {h−lj+1+g+ft, li ′} = {χ, li ′} = 0, hence {f+, li ′} = 0 for i′ ∈ {1, . . ., j}.
To conclude, using our hypothesis Tj+1˜ε ·< r ′ ·< s ′ , as in Lemma 3.17 we can
show that our transformation ϕj+1 maps Drj+1−r ′ ,sj+1−s ′(ωj+1) into Drj+1,sj+1 (ωj+1) with
|ϕj+1 − Id|rj+1−r ′ ,sj+1−s ′

3.A - Proof of the normal form 89
Proof of Proposition 3.6. The proof is by induction on j ∈ {1, . . .,n}.
First step. Here we assume ( Ã1) and we will apply m times our first iterative
Lemma 3.17, starting with the Hamiltonian
H 0 = H = h + g 0 + f 0
where g 0 = 0 and f 0 = f and choosing uniformly at each step
r ′ = (3m) −1 r1, s ′ = (3m) −1 s1.
Since m ≥ 1, we have 0 < r ′ < r1, 0 < s ′ < s1 and using ( Ã1), we have
T1˜ε < r ′ < s ′ ,
so that the lemma can indeed be applied at each step. For i ∈ {0, . . ., m − 1}, the
Hamiltonian H i = h + g i + f i at step i is transformed into
H i+1 = H i ◦ ϕ i 1 = h + gi+1 + f i+1 .
For each i ∈ {0, . . ., m}, we obviously have {g i , l1} = 0 and we claim that the estimates
||X g i|| r i 1 ,s i 1

90 Generic exponential stability without small divisors
assume that the estimates (29) are satisfied for each k ≤ i, where i ∈ {0, . . .,m − 1}.
For k ∈ {0, . . ., i}, since ||X f k|| r k 1 ,s k 1 < ˜εk we get that
and therefore
||Xgk+1 − Xgk|| k+1
r1 ,s k+1 < ˜εk,
1
||X g i+1|| r i+1
1 ,s i+1
1 ≤
i
˜εk

3.B - SDM functions 91
with g 0 j = 0 and f0 j = gj, we can apply m times our second iterative Lemma 3.18 to
have the following: there exists an analytic symplectic transformation
Φj+1 : D2rj+1/3,2sj+1/3(ωj+1) → Drj+1,sj+1 (ωj+1)
of the form Φj+1 = ϕ 0 j+1 ◦ · · · ◦ ϕ m−1
j+1 such that |Φj+1 − Id|2rj+1/3,2sj+1/3 ·< rj+1 and
(h + gj) ◦ Φj+1 = h + g m j + f m j ,
with {g m j , li} = 0 for i ∈ {1, . . ., j + 1}, and the estimates
Now we set
||Xg m j ||2rj+1/3,2sj+1/3

92 Generic exponential stability without small divisors
3.B.1 Steepness
3.B.1.1. We denote by GAB(n, k) the set of all affine subspaces of R n of dimension k
intersecting the ball B, and by GA L B (n, k) those subspaces with direction in GL (n, k) (the
latter is the space of linear subspaces of R n of dimension k whose orthogonal complement
is spanned by integer vectors of length less than or equal to L). Let us recall the classical
steepness condition, originally introduced by N.N. Nekhoroshev ([Nek77]).
Definition 3.19. A function h ∈ C 2 (B) is said to be steep if it has no critical points
and if for any k ∈ {1, . . ., n − 1}, there exist an index pk > 0 and coefficients Ck > 0,
δk > 0 such that for any affine subspace λk ∈ GAB(n, k) and any continuous curve Γ
from [0, 1] to λk ∩ B with
Γ(0) − Γ(1) = r < δk,
there exists t∗ ∈ [0, 1] such that:
Γ(t) − Γ(0) < r, t ∈ [0, t∗],
ΠΛk (∇h(Γ(t∗))) > Ckr pk
where ΠΛk is the projection onto Λk, the direction of λk.
The function is said to be symmetrically steep (or shortly S-steep) if the above property
is also satisfied for k = n, with an index pn > 0 and coefficients Cn > 0, δn > 0.
Let us remark that S-steep functions are allowed to have critical points. Those
definitions are rather obscure, but in fact it can be given a simpler and more geometric
interpretation, as was shown by Ilyashenko ([Ily86]) and Niederman ([Nie06]). Important
examples of steep functions are given by the class of strictly convex (or quasi-convex)
functions, with all the steepness indices equal to one.
3.B.1.2. A typical example of non-steep function, which is due to Nekhoroshev, is
h(I1, I2) = I 2 1 − I2 2 , and it is not exponentially stable: for the perturbation hε(I1, I2) =
I 2 1 − I2 2 + ε sin(I1 + I2), any solution with I1(0) = I2(0) has a fast drift, that is a drift
of order one on a time scale of order ε−1 (this is obviously the fastest drift possible).
) we recover
But adding a third order term in the previous example (for example I3 2
steepness, and this is in fact a general phenomenon. Indeed, non-steep functions has
infinite codimension among smooth functions, or more precisely, if Jr(n) is the space of
r-jets of C∞ functions on an open set of Rn , then Nekhoroshev proved in [Nek79] that the
set of r-jets of non-steep functions is an algebraic subset of Jr(n) which codimension goes
to infinity has r goes to infinity. In this sense, steep functions are "generic". However,
for n ≥ 3, a quadratic Hamiltonian is steep only if it is sign definite, which is a strong
assumption, and more generally a polynomial is generically steep only if its degree is
sufficiently high (of order n2 if n is the number of degrees of freedom). Hence polynomials
of lower degree are generically non-steep (see [LM88]). This is clearly a shortcoming,
and we will see at the end of the next section the advantage of our genericity condition.
3.B.1.3. Steepness (or S-steepness) is a sufficient condition to ensure exponential
stability, but this is not necessary, as was first noticed by Morbidelli and Guzzo (see
[MG96]). They considered the Hamiltonian h(I1, I2) = I2 1 − αI2 2 , which is non-steep for

3.B - SDM functions 93
any value of α > 0, and noticed that a "fast drift" is not possible if √ α is "strongly"
irrational. Therefore a Diophantine condition on √ α should ensure exponential stability.
Such considerations were then generalized by Niederman who introduced the class
of "Diophantine Morse" functions and who proved that they are exponentially stable
([Nie07b]). The only difference between these functions and the "Simultaneous Diophantine
Morse" functions we use in this chapter is that Diophantine Morse functions
consider subspaces in GL(n, k), which are generated by integer vectors of length bounded
by L, while here we are looking at subspaces in G L (n, k) where the latter condition is
imposed on the orthogonal complement. This reflects the difference between the method
of proof: in ([Nie07b]) the analytic part was based on classical small divisors techniques
(that is linear Diophantine approximation) and therefore required an adapted geometric
assumption, while here we simply rely on the most basic theorem of simultaneous Diophantine
approximation (and this explains the name Simultaneous Diophantine Morse
functions).
3.B.1.4. In both cases, the use of such a class of functions has two advantages. The
first one is that these functions are generic in a much more clearer sense than steep
functions, and this will be explained in the next section. The second advantage is that
they are in some sense more general than the usual steep functions, since we only have
to consider curves in some specific affine subspaces. This is explained in the proposition
below.
Proposition 3.7. Let h ∈ SDM τ γ (B), assume that |h| C3 (B) < M and take r < 1. Then
for any affine subspace λ ∈ GAL B (n, k) and any continuous curve Γ from [0, 1] to Λ ∩ B
with
Γ(0) − Γ(1) = r < (2M) −1 γL −τ ,
there exists t∗ ∈ [0, 1] such that:
Γ(t) − Γ(0) ≤ r, t ∈ [0, t∗],
ΠΛ(∇h(Γ(t∗))) > 1
2 r2
where ΠΛ is the projection onto Λ, the direction of λ.
Proof. It is enough to check that these properties are satisfied for a vector space Λ ∈
GL (n, k), since any affine subspace λ ∈ GAL B (n, k) is of the form λ = w + Λ with
Λ ∈ GL (n, k) for some vector w. So consider a continuous curve Γ from [0, 1] to Λ ∩ B
with length r < 1 satisfying
Γ(0) − Γ(1) = r < (2M) −1 γL −τ .
We will denote by (u(t), v) the coordinates of Γ(t) for t ∈ [0, 1] in a basis adapted to the
orthogonal decomposition Λ ⊕ Λ ⊥ . Therefore
ΠΛ(∇h(Γ(t))) = ∂uhΛ(u(t), v)
for all t ∈ [0, 1]. We will distinguish distinguish two cases.
For the first one, we suppose that
∂uhΛ(u(0), v) > 2 −1 r 2 ,

94 Generic exponential stability without small divisors
so the conclusion trivially holds for t∗ = 0.
For the second one, we have
but since r 2 < r < γL −τ , this gives
∂uhΛ(u(0), v) ≤ 2 −1 r 2 , (30)
∂uhΛ(u(0), v) ≤ γL −τ .
Now h ∈ SDM τ γ (B), so we can apply the definition at the point (u(0), v), and for any
η ∈ Rk \ {0} we obtain
∂uuhΛ(u(0), v).η > γL −τ η. (31)
Take any ũ such that ũ − u(0) < (2M) −1 γL −τ . We can apply Taylor formula with
integral remainder to obtain
∂uhΛ(ũ, v) − ∂uhΛ(u(0), v) =
1
Now since M bounds the third derivative of h, we have
0
∂uuhΛ(u(0) + t(ũ − u(0)), v).(ũ − u(0))dt.
∂uuhΛ(u(0) + t(ũ − u(0)), v) − ∂uuhΛ(u(0), v) ≤ Mtũ − u(0) ≤ 2 −1 γL −τ t,
and this yields
∂uhΛ(ũ, v) − ∂uhΛ(u(0), v) ≥ ∂uuhΛ(u(0), v).(ũ − u(0))
which in turns, using (31) with η = ũ − u(0), gives
Now we define
∂uhΛ(ũ, v) − ∂uhΛ(u(0), v) ≥
so trivially we have
Furthermore, we have
−2 −1 γL −τ
1
γL −τ − 2 −1 γL −τ
0
tũ − u(0)dt,
1
tdt ũ − u(0)
0
≥ 2 −1 γL −τ ũ − u(0). (32)
t∗ = inf {Γ(t) − Γ(0) = r},
t∈[0,1]
Γ(t) − Γ(0) ≤ r, t ∈ [0, t∗].
∂uhΛ(u(t∗), v) ≥ ∂uhΛ(u(t∗), v) − ∂uhΛ(u(0), v) − ∂uhΛ(u(0), v),
and so using (30), (32) and recalling that u(t∗) − u(0) = r and γL −τ > 2r we obtain
and this is the desired estimate.
∂uhΛ(u(t∗), v) ≥ 2 −1 γL −τ r − 2 −1 r 2
> r 2 − 2 −1 r 2
= 2 −1 r 2 ,

3.B - SDM functions 95
3.B.2 Prevalence
3.B.2.1. Here we will prove our results of genericity concerning SDM functions, that is
Theorem 3.4 and Corollary 3.5. Our main tool is the following lemma, which is proved
in [Nie07b] and relies on the quantitative Morse-Sard theory developed by Yomdin (see
[YC04] and [Yom83]).
Lemma 3.20. Let κ ∈]0, 1[ and g ∈ C2n+1 (B, Rk ). There exist a positive constant ck
and a subset Cκ ⊆ Rk such that
√
λk(Cκ) ≤ ck κ,
and for any ζ /∈ Cκ, the function g ζ defined by g ζ (x) = g(x) − ζ satisfies the following:
for any x ∈ B,
g ζ (x) ≤ κ =⇒ dg ζ (x).ν > κν,
for any ν ∈ R n \ {0}.
In the above statement, the set Cκ is a "nearly-critical set" for the function g.
3.B.2.2. Let us prove Theorem 3.4.
Proof of Theorem 3.4. Recall that we are given a function h ∈ C 2n+2 (B). The proof is
divided in two steps: first, we will describe the set of parameters ξ ∈ R n for which the
function hξ, defined by hξ(I) = h(I) − ξ.I, is not in SDM τ (B), and then, in a second
step, we will show that this set has zero Lebesgue measure, for τ > 2(n 2 + 1). In the
sequel, given k ∈ {1, . . ., n}, we denote by λk the Lebesgue measure of R k .
First step. Given an element Λ ∈ G L (n, k), let ΠΛ the projection onto this subspace
and consider the associate function hΛ (recall that hΛ is just the function h written
in coordinates adapted to the orthogonal decomposition Λ ⊕ Λ ⊥ ). Let us define the
function
g = ∂uhΛ,
which belongs C 2n+1 (B, R k ), and apply to this function Lemma 3.20 with the value
κ = γL −τ . We find a "nearly-critical" set Cκ = Cγ,τ,L ⊆ R k with the measure estimate
such that for any ζ /∈ Cγ,τ,L and any (u, v) ∈ B,
for any ν ∈ R n \ {0}.
λk(Cγ,τ,L) ≤ ckγ 1 τ
2L
− 2, (33)
g ζ (u, v) ≤ κ =⇒ dg ζ (u, v).ν > κν, (34)
Now take any ζ /∈ Cγ,τ,L, any ξ ∈ Π −1
Λ (ζ) and consider the modified function hξ as
well as its version hξ,Λ. Since
∂uhξ,Λ = ∂uhΛ − ζ = g − ζ = g ζ ,
and ∂uuhξ,Λ = ∂uuhΛ is just some restriction of dg, the estimate (34) gives for any
(u, v) ∈ B,
∂uhξ,Λ(u, v) ≤ γL −τ =⇒ ∂uuhξ,Λ(u, v).η > γL −τ η (35)

96 Generic exponential stability without small divisors
for any η ∈ R k \ {0}. So let Cγ,τ,L,Λ = Π −1
Λ (Cγ,τ,L), and define
Cγ,τ =
L∈N∗ k∈{1,...,n} Λ∈GL (n,k)
Cγ,τ,L,Λ.
As a consequence of the estimate (35), the function hξ ∈ SDM τ γ (B) provided that
ξ /∈ Cγ,τ, hence hξ ∈ SDM τ (B) provided that ξ /∈ Cτ, where
Cτ =
γ>0
Second step. It remains to prove that Cτ has zero Lebesgue measure under our
assumption that τ > 2(n 2 + 1). For an integer m ∈ N ∗ , we define C m γ,τ,L,Λ (resp. Cm γ,τ
and C m τ ) as the intersection of Cγ,τ,L,Λ (resp. Cγ,τ and Cτ) with the ball of R n of radius
m centered at the origin. As a consequence of (33) and Fubini-Tonelli theorem, one has
Cγ,τ.
λn(C m γ,τ,L,Λ) ≤ Vn,mckγ 1 τ
2L
− 2
where Vn,m = mnπn/2Γ(n/2 + 1) −1 is the volume of the ball of Rn of radius m centered
at the origin. Therefore
⎛
⎝
⎞
⎠ ≤ |G L τ
−
(n, k)|Vn,mckL 2γ 1
2,
λn
Λ∈G L (n,k)
C m γ,τ,L,Λ
with |G L (n, k)| the cardinal of G L (n, k). But obviously |G L (n, k)| ≤ L n2
Now
and so
λn
⎛
λn
⎝
⎛
⎝
Λ∈G L (n,k)
k∈{1,...,n} Λ∈GL (n,k)
λn(C m γ,τ ) ≤ Vn,m
⎞
C m ⎠
γ,τ,L,Λ ≤ Vn,mckL n2− τ
2γ 1
C m γ,τ,L,Λ
n
k=1
⎞
⎠ ≤ Vn,m
ck
+∞
L=1
n
k=1
ck
L n2 − τ
2
2.
L n2− τ
2γ 1
2,
γ 1
2
and hence
where the sum in the right-hand side of the last estimate is finite since we are assuming
τ > 2(n 2 + 1). This shows that
and as Cτ =
m≥1 Cm τ
and this concludes the proof.
λn(C m τ ) = inf
γ>0 λn(C m γ,τ) = 0,
we finally obtain
λn(Cτ) = 0,

3.B - SDM functions 97
3.B.2.3. As we mentioned in the introduction, there is a notion of genericity in infinite
dimensional vector spaces called prevalence, first introduced in a different setting by
Christensen ([Chr73]) and rediscovered by Hunt, Sauer and Yorke ([HSY92], see also
[KH08] and [OY05]).
Definition 3.21. Let E be a completely metrizable topological vector space. A Borel
subset S ⊆ E is said to be shy if there exists a Borel measure µ on E, with 0 < µ(C) < ∞
for some compact set C ⊆ E, and such that µ(x + S) = 0 for all x ∈ E.
An arbitrary set is called shy if it is contained in a shy Borel subset, and finally the
complement of a shy set is called prevalent.
The following "genericity" properties are easy to check ([OY05]): a prevalent set is
dense, a set containing a prevalent set is also prevalent, and prevalent sets are stable
under translation and countable intersection.
Furthermore, we have an easy but useful criterion for a set to be prevalent.
Proposition 3.8 ([HSY92]). Let S be a subset of E. Suppose there exists a finitedimensional
subspace F of E such that x + S has full λF-measure for all x ∈ E. Then
S is prevalent.
3.B.2.4. Now we can prove our Corollary 3.5.
Proof of Corollary 3.5. Let E = C 2n+2 (B), S = SDM τ (B) for τ > 2(n 2 +1) and F the
space of linear forms of R n restricted to B. Then F is a linear subspace of C 2n+2 (B) of
dimension n, and the conclusion follows immediately from Theorem 3.4 and the above
Proposition 3.8.
3.B.2.5. To conclude, let us compare our generic condition with the usual steepness
property. First, our condition is prevalent in the space C k (B), with k ≥ 2n+2, and this
is not true for steep functions. But more importantly, as prevalence is nothing but "full
Lebesgue measure" in finite dimension, given any non zero integers m and n, Lebesgue
almost all polynomial Hamiltonian hm of degree m with n degrees of freedom is SDM,
but not steep unless m is of order n 2 . This remark turns out to be very useful when
studying the stability of invariant tori under generic conditions (see the next chapter).

98 Generic exponential stability without small divisors

4.1 - Introduction and main results 99
4 Generic super-exponential stability for invariant
tori
In this chapter, we consider solutions starting close to some linearly stable invariant
tori in an analytic Hamiltonian system and we prove results of stickiness, that is of
stability for a super-exponentially long interval of time, under generic conditions. The
proof combines classical Birkhoff normal forms and the new method to obtain generic
Nekhoroshev’s estimates developed in the first chapter. We will mainly focus on the
neighbourhood of elliptic fixed points, since with our approach the other cases are completely
similar. This chapter is based on [Bou09b].
4.1 Introduction and main results
In this chapter, we are interested in the stability properties of some linearly stable
invariant tori in analytic Hamiltonian systems. Let us begin by the case of elliptic fixed
points.
4.1.0.1. As the problem is local, it is enough to consider a Hamiltonian H defined and
analytic on an open neighbourhood of 0 in R 2n , having the origin as a fixed point. Up
to an irrelevant additive constant and expanding the Hamiltonian as a power series at
the origin, we can write
H(z) = H2(z) + V (z),
where z is sufficiently close to 0 in R 2n , H2 is the quadratic part of H at 0 and V (z) =
O(||z|| 3 ). Recall that the fixed point is said to be elliptic if the spectrum of the linearized
system is purely imaginary. Then it has the form {±iα1, . . .,±iαn}, for some vector
α = (α1, . . .,αn) ∈ R n which is called the normal (or characteristic) frequency. Due to
the symplectic character of the equations, such equilibria are the only linearly stable
fixed points. Now we assume that the components of α are all distinct so that we can
make a symplectic linear change of variables that diagonalizes the quadratic part:
H(z) =
n
i=1
αi
2 (z2 i + z 2 n+i) + V (z) = α. Ĩ + V (z),
where Ĩ = Ĩ(z) is the vector of "formal action", that is
Ĩ(z) = 1
2 (z2 1 + z 2 n+1, . . .,z 2 n + z 2 2n) ∈ R n .
Assuming the components of α are all of the same sign, it is easy to see that H is a
Lyapunov function so the fixed point is stable. But in the general case, one has to study
the influence of the higher order terms V (z), and we will explain how it can be done
using classical perturbation theory.
4.1.0.2. Let us first note that, given a solution z(t) of H, if Ĩ(t) = Ĩ(z(t)) then
| Ĩ(t)|1 =
n
| Ĩi(t)|
i=1

100 Generic super-exponential stability for invariant tori
is (up to a factor one-half) the square of the Euclidean distance of z(t) to the origin, so
that Lyapunov stability can be proved if | Ĩ(t) − Ĩ(0)|1 does not vary much for all times.
Now in order to study the dynamics on a small neighbourhood of size ρ around the
origin in R 2n , it is more convenient to change coordinates by performing the standard
scalings
z ↦−→ ρz, H ↦−→ ρ −2 H,
to have a Hamiltonian defined on a fixed neighbourhood of zero in R 2n . Then, by
analyticity, we extend the resulting Hamiltonian to a holomorphic function on some
complex neighbourhood of zero in C 2n . So eventually we will consider the following
setting: we define the Euclidean ball in C 2n
Ds = {z ∈ C 2n | ||z|| < s}
of radius s around the origin, and if As is the space of holomorphic functions on Ds
which are real valued for real arguments, endowed with its usual supremum norm | . |s,
we consider
H(z) = α. Ĩ + f(z)
(A)
H ∈ As, |f|s < ρ.
Let us emphasize that the small parameter ρ, which was originally describing the size
of the neighbourhood of 0, now describes the size of the "perturbation" f on a neighbourhood
of fixed size s. Without loss of generality, we may assume s > 3.
4.1.0.3. Probably the main tool to investigate stability properties is the construction of
normal forms using averaging methods, and in this case these are the so-called Birkhoff
normal forms. For an integer m ≥ 1, assuming α is non-resonant up to order 2m, that
is
k.α = 0, k ∈ Z n , 0 < |k|1 ≤ 2m,
there exists an analytic symplectic transformation Φm close to identity such that H ◦Φm
is in Birkhoff normal form up to order 2m, that is
H ◦ Φm(z) = hm( Ĩ) + fm(z),
where hm is a polynomial of degree at most m in the Ĩ variables, and the remainder fm
is roughly of order ρ 2m−1 (since before the scaling fm(z) is of order z 2m+1 , see [Bir66],
or [Dou88] for a more recent exposition). The polynomials hm are uniquely defined once
α is fixed, and are usually called the Birkhoff invariants. Therefore the transformed
Hamiltonian is the sum of an integrable part hm, for which the origin is trivially stable,
since Ĩ(t) is constant for all times, and a smaller perturbation fm. Moreover, if α is
non-resonant up to any order, we can even define a formal symplectic transformation
Φ∞ and a formal power series h∞ =
k≥1 hk , with hm = m
k=1 hk , such that
H ◦ Φ∞(z) = h∞( Ĩ).
In general the series h∞ is divergent (this is a result of Siegel) and the convergence properties
of the transformation Φ∞ are even more subtle (see [PM03]). However, Birkhoff
normal forms at finite order are still very useful, not only because the "perturbation" fm
is made smaller, but also because the "integrable" part hm, for m ≥ 2, is now non-linear
and other classical techniques from perturbation theory can be used.

4.1 - Introduction and main results 101
4.1.0.4. First, in the case n = 2, a complete result of stability follows from KAM
theory. Indeed, if the frequency α ∈ R 2 is non-resonant up to order 4, the Birkhoff
normal form reads
H(z) = α. Ĩ + βĨ.Ĩ + f2(z),
with β a symmetric matrix of size n = 2 and f2 a small perturbation. This time we
consider the non-linear part h2( Ĩ) = α.Ĩ + βĨ.Ĩ as the integrable system, and if it is
isoenergetically non degenerate, the persistence of two-dimensional tori in each energy
level close to the fixed point implies Lyapunov stability (see [AKN06], [Arn63b], or
[Arn61] for other results).
However, for n ≥ 3, it is believed that "generic" elliptic fixed points are unstable,
although this is totally unclear for the moment (see [DLC83], [Dou88] and [KMV04]).
Therefore, for n ≥ 3, stability results under general assumptions can only concern
finite but hopefully long intervals of time, and this is the content of the chapter. More
precisely, we will prove, under generic assumptions and provided ρ is sufficiently small,
that for all initial conditions the variation | Ĩ(t) −Ĩ(0)|1 is of order ρ for t ∈ T(ρ), where
T(ρ) is an interval of time of order exp (exp(ρ −1 )) (see Theorem 4.1 for a precise formulation).
Such a phenomenon of super-stability is sometimes called "stickiness" ([PW94]).
The interpretation in the original coordinates is the following: if a solution starts in
a sufficiently small neighbourhood of the origin, it stays in some larger neighbourhood
during an interval of time which is super-exponentially long with respect to the inverse
of the initial distance to the origin. But first, let us describe previously known results
on exponential stability, where there were basically two strategies.
4.1.0.5. In a first approach, one assumes a Diophantine condition on α, that is there
exist γ > 0 and τ > n − 1 such that
|k.α| ≥ γ|k| −τ
1 , k ∈ Z n \ {0},
but no conditions on the Birkhoff invariants. From the point of view of perturbation
theory, the linear part is considered as the integrable system. In particular, α is
non-resonant up to any order, hence we can perform any finite number of Birkhoff normalizations,
and since we have a control on the small divisors, we can precisely estimate
the size of the remainder fm (in terms of γ and τ). The usual trick is then to optimize
the choice of m as a function of ρ in order to obtain an exponentially small remainder
with respect to ρ −1 . Therefore the exponential stability is immediately read from the
normal form, and this requires only an assumption on the linear part (see [GDF + 89] or
[DG96b]). The above Diophantine condition has full Lebesgue measure. However, as we
will see later, the threshold of the perturbation and the constants of stability are very
sensitive to the Diophantine properties of α, in particular the small parameter γ.
4.1.0.6. The second approach is fundamentally different, and it does not rely on the
arithmetic properties of α. Here, one just assumes that α is non-resonant up to order
4, so that the Hamiltonian reduces to
H(z) = α. Ĩ + βĨ.Ĩ + f2(z).
In this case h2( Ĩ) = α.Ĩ + βĨ.Ĩ is considered as the integrable system (β being a symmetric
matrix of size n). Now we suppose that the non-linear part is convex, which

102 Generic super-exponential stability for invariant tori
is equivalent to β being sign definite. Under those assumptions, it was predicted and
partially proved by Lochak ([Loc92] and [Loc95]), and completely proved independently
by Niederman ([Nie98]) and Fassò, Guzzo and Benettin ([FGB98] and [GFB98]) that
exponential stability holds. Their proofs are based on the implementation of Nekhoroshev’s
estimates in Cartesian coordinates, but they are radically different: the first one
uses Lochak’s method of periodic averaging and simultaneous Diophantine approximations,
while the second one is based on Nekhoroshev’s original mechanism. The proof
of Niederman was later clarified by Pöschel ([Pös99b]). However, the method of Lochak
was restricted to the convex case, and it was not clear how to remove this hypothesis to
have a result valid in a more general context.
4.1.0.7. Here, using the method we introduced in the first chapter, we are able to
replace the convexity condition by a generic assumption. Then, combining both Birkhoff
theory and Nekhoroshev theory as in [MG95], we will obtain the following result.
Theorem 4.1. Suppose H is as in (A), with α non-resonant up to any order. Then
under a generic condition (G) on h∞, there exist positive constants a, a ′ , c1, c2 and ρ0
such that for ρ ≤ ρ0, every solution z(t) of H with | Ĩ(0)|1 < 1 satisfies
| Ĩ(t) − Ĩ(0)|1
< c1ρ, |t| < exp ρ −a′
exp(c2a ′ ρ −a
) .
Denoting h∞ =
k≥1 hk and hm = m k=1 hk , let us explain our generic condition (G)
on the formal power series h∞. In fact
(G) =
(Gm)
m∈N ∗
consists in countably many conditions, where (Gm) is a condition on hm. The first
condition (G1) requires that h1(I) = α.I with a (γ, τ)-Diophantine vector α. The other
conditions (Gm), for m ≥ 2, are that each polynomial function hm belongs to a special
class of functions called SDM
τ ′
γ
′ which was introduced in the first chapter (SDM stands
for "Simultaneous Diophantine Morse" functions, see Appendix 4.A for a definition). In
this appendix we will show that each condition (Gm) is of full Lebesgue measure in the
finite dimensional space of polynomials of degree m with n variables, assuming τ and
τ ′ are large enough. This is well-known for m = 1, it will be elementary for m = 2 (see
Theorem 4.11) but for m > 2 it requires the quantitative Morse-Sard theory of Yomdin
([Yom83], [YC04], see Proposition 4.3 in the appendix). Let us point out that this would
have not been possible if we had assumed hm, for m ≥ 2, to be steep in the sense of
Nekhoroshev, as polynomials are generically steep only if their degrees are sufficiently
large with respect to the number of degrees of freedom (see [LM88]).
Our condition (G) on the formal series h∞ is therefore of "full Lebesgue measure
at any order". From an abstract point of view, this condition defines a prevalent set
in the space of formal power series, where prevalence is an analog of the notion of full
Lebesgue measure in the context of infinite dimensional vector spaces. This will be
proved in Appendix 4.A, Theorem 4.9. In Theorem 4.1, we can choose the exponents
a = (1 + τ) −1 , a ′ = 3 −1 (2(n + 1)τ ′ ) −n ,
and our threshold ρ0 depends in particular on γ and γ ′ . Moreover our constants c1 and
c2 also depend on γ but not on γ ′ , and we shall be a little more precise later on.

4.1 - Introduction and main results 103
As we have already explained, the proof is based on a combination of Birkhoff normalizations
up to an exponentially small remainder, which are well-known (a statement
is recalled in Proposition 4.1 below), and Nekhoroshev’s estimates for a generic integrable
Hamiltonian near an elliptic fixed point (Theorem 4.3 below). The latter result
is new, and it will follow rather easily from the new approach of Nekhoroshev theory in
a generic case taken in the first chapter.
4.1.0.8. As a direct consequence of our Nekhoroshev’s estimates near an elliptic fixed
point, we can derive an exponential stability result more general than those obtained in
[FGB98] and [Nie98]. Like in those papers, we only require α to be non-resonant up to
order 4, and after the scalings
z ↦−→ ρz, H ↦−→ ρ −4 H, α ↦−→ ρ 2 α,
we consider
H(z) = α. Ĩ + βĨ.Ĩ + f(z)
H ∈ As, |f|s < ρ.
However, instead of assuming that β is sign definite, our result applies to Lebesgue
almost all symmetric matrices β without any condition on α. Let Sn(R) be the space
of symmetric matrices of size n with real entries.
Theorem 4.2. Suppose H is as in (B). For Lebesgue almost all β ∈ Sn(R), there
exist positive constants a ′ , b ′ and ρ0 such that, for ρ ≤ ρ0, every solution z(t) of H with
| Ĩ(0)|1 < 1 satisfies
| Ĩ(t) − Ĩ(0)|1 < n(n 2 + 1)ρ −b′
, |t| < exp(ρ −a′
).
The above theorem is a direct consequence of Theorem 4.3 below, provided that
′
h2( Ĩ) = α.Ĩ + βĨ.Ĩ belongs to SDMτ γ ′ . But we will prove in Appendix 4.A that this
happens for almost all symmetric matrices β, independently of α (see Theorem 4.11).
Once again, let us also mention that this result is not possible in the steep case, as
the quadratic part h2 ( Ĩ) = βĨ.Ĩ is steep if and only if β is sign definite. In the above
statement one can choose the exponents
and the threshold ρ0 depends on γ ′ .
a ′ = b ′ = 3 −1 (2(n + 1)τ ′ ) −n ,
4.1.0.9. Let us add that in order to avoid useless expressions, we will only keep track
of the small parameters ρ, γ and γ ′ and replace any other positive constants by a dot
(·) when it is convenient.
Moreover, in this text we shall use various norms for vectors v ∈ R n or v ∈ C n : | . |
will be the supremum norm, | . |1 the ℓ1-norm and . the Euclidean (or Hermitian)
norm.
4.1.0.10. Let us now describe the plan of this chapter. Section 4.2 is devoted to the
proof of Theorem 4.1 and Theorem 4.2. In 4.2.1, we give a statement of the Birkhoff
normal form up to an exponentially small remainder. In 4.2.2, we will explain how
Nekhoroshev’s estimates obtained in the first chapter generalize in the neighbourhood
(B)

104 Generic super-exponential stability for invariant tori
of elliptic fixed points, and how they imply Theorem 4.2. In 4.2.3, we will show how
Theorem 4.1 follows from a simple combination of Birkhoff’s estimates and Nekhoroshev’s
estimates, provided our assumption on h∞ is satisfied. Then, in section 4.3, we
will state similar results for invariant Lagrangian tori and more generally for invariant
linearly stable isotropic reducible tori. Finally, an appendix is devoted to our genericity
assumptions.
4.2 Proof of Theorem 4.1 and Theorem 4.2
In the sequel, we recall that we will use the "formal" actions
Ĩ =
Ĩ(z) = 1
2 (z2 1 + z2 n+1 , . . ., z2 n + z2 2n ) ∈ Rn ,
but one has to remember that these are nothing but notations for expressions in z ∈ R 2n .
Moreover, we will also need to use complex coordinates for the normal forms, and,
abusing notations, we will also denote them by z ∈ C 2n , but of course the solutions we
consider are real.
4.2.1 Birkhoff’s estimates
Here we consider a Hamiltonian as in (A), and we assume that the vector α is (γ, τ)-
Diophantine. In this context, the following result is classical.
Proposition 4.1. Under the previous assumptions, if ρ

4.2 - Proof of Theorem 4.1 and Theorem 4.2 105
4.2.2 Nekhoroshev’s estimates and proof of Theorem 4.2
4.2.2.1. Here we consider the Hamiltonian
H(z) = h( Ĩ) + f(z)
τ ′
H ∈ As, h ∈ SDMγ ′ , |f|s < ε
and we have assumed that, on the real part of the domain, the derivatives up to order 3
τ ′
of h are uniformly bounded by some constant M > 1. The definition of the set SDMγ ′
is recalled in Appendix 4.A.
Theorem 4.3. Let H be as in (E), with τ ′ ≥ 2 and γ ′ ≤ 1. Then there exists ε0 such
that if ε ≤ ε0, for every solution z(t) with | Ĩ(0)| < 1, we have
| Ĩ(t) − Ĩ(0)| < (n2 + 1)ε b′
, |t| < exp(ε −a′
),
with the exponents a ′ = b ′ = 3 −1 (2(n + 1)τ ′ ) −n .
Theorem 4.2 is now an immediate consequence of this result and Theorem 4.11 (see
Appendix 4.A).
Proof of Theorem 4.2. From Theorem 4.11, we know that for almost all β ∈ Sn(R), the
′
Hamiltonian h( Ĩ) = α.Ĩ +βĨ.Ĩ belongs to SDMτ (B) with τ ′ > n2 +1. So we can apply
Theorem 4.3: for every solution z(t) with | Ĩ(0)| < 1, we have
| Ĩ(t) − Ĩ(0)| < (n2 + 1)ε b′
, |t| < exp(ε −a′
),
with the exponents a ′ = b ′ = 3 −1 (2(n + 1)τ ′ ) −n . In particular, this gives
for every solution z(t) with | Ĩ(0)|1 < 1.
| Ĩ(t) − Ĩ(0)|1 < n(n 2 + 1)ε b′
, |t| < exp(ε −a′
),
4.2.2.2. The statement of Theorem 4.3 is the analogue of Theorem 3.6. However,
the difference is that here we are using Cartesian coordinates and not action-angle
coordinates (i.e. symplectic polar coordinates), and we cannot use the latter since they
become singular at the origin. So we cannot apply directly Theorem 3.6. This is not
a serious issue when applying KAM theory in this context (see [Arn63b] or [Pös82] for
example), but this becomes problematic in Nekhoroshev theory (see [Loc92] or [Loc95]
for detailed explanations). This result was only conjectured by Nekhoroshev in [Nek77],
and it took a long time before it could be solved in the convex case ([Nie98],[FGB98]).
Here we are able to solve this problem in the generic case. The reason is that even
though we cannot apply the result of the first chapter, we can use exactly the same
approach, since the method of averaging along unperturbed periodic flows is intrinsic,
i.e. independent of the choice of coordinates, a fact that was first used implicitly in
[Nie98] and made completely clear in [Pös99b].
The proof of such estimates usually requires an analytic part, which boils down to
the construction of suitable normal forms, and a geometric part. The geometric part of
(E)

106 Generic super-exponential stability for invariant tori
the first chapter goes exactly the same way, so in the sequel we will restrict ourselves to
indicating the very slight modifications in the construction of the normal forms.
4.2.2.3. Consider linearly independent periodic vectors ω1, . . .,ωn of R n , with periods
(T1, . . .,Tn), that is
Define the domains
Tj = inf{t > 0 | tωi ∈ Z n }, 1 ≤ j ≤ n.
Drj,sj (ωj) = {z ∈ Dsj | |∇h(Ĩ) − ωj|

4.2 - Proof of Theorem 4.1 and Theorem 4.2 107
The proof is completely analogous to the proof of the Proposition 3.6, Appendix 3.A,
to which we refer for more details. In fact, here the proof is even simpler since one does
not have to use "weighted" norms for vector fields. It relies on a finite composition of
averaging along the periodic flows generated by lj, j ∈ {1, . . ., n}. The case j = 1 is due
to Pöschel ([Pös99b]) and, for j > 1, the proof goes by induction using our assumption
(Aj), j ∈ {1, . . ., n}.
Once we have this normal form, the rest of the proof of Theorem 3.6 goes exactly
the same way: every solution z(t) of H with | Ĩ(0)| < 1 satisfies
| Ĩ(t) − Ĩ(0)| < (n2 + 1)ε b′
, |t| < exp(ε −a′
),
provided that ε ≤ ε0, with ε0 depending on n, s, M, γ ′ and τ ′ and with the exponents
4.2.3 Proof of Theorem 4.1
a ′ = b ′ = 3 −1 (2(n + 1)τ ′ ) −n .
Now we can finally prove Theorem 4.1, by using successively Birkhoff’s estimates and
Nekhoroshev’s estimates.
Proof of Theorem 4.1. Let H be as in (A), first assume that ρ < ρ1 with ρ1 =·γ so
that using our assumption (G1) we can apply Proposition 4.1: there exist an integer
m = m(ρ) and an analytic symplectic transformation
such that
Φm : D3s/4 → Ds
H ◦ Φm(z) = hm( Ĩ) + fm(z)
is in Birkhoff normal form, with a remainder fm satisfying the estimate
|fm|3s/4

108 Generic super-exponential stability for invariant tori
However, one has
ρ b′
exp −b ′ (γρ −1 ) a < γ −1 ρ,
and as Φm satisfies |Φm − Id|3s/4

4.3 - Further results and comments 109
However, it is important to note that one cannot obtain a statement similar to
Theorem 4.2, simply because in this case a non-resonant condition up to a finite order
does not allow to build the corresponding Birkhoff normal form.
If we compare this result with [MG95], our assumption is generic and we do not
require any convexity. But of course the price to pay is that one has to consider the full
set of Birkhoff invariants.
4.3.0.2. As a final result, one can also obtain similar estimates for the general case of
a linearly stable lower-dimensional torus, under the common assumptions of isotropicity
and reducibility (which were automatic for a fixed point or a Lagrangian torus). In that
context, it is enough to consider a Hamiltonian defined in Tk ×Rk ×R2l (by isotropicity),
of the form
H(θ, I, z) = ω.I + 1
Bz.z + F(θ, I, z).
2
Here B is a symmetric matrix (constant by reducibility) such that J2lB has a
purely imaginary spectrum (J2l being the canonical symplectic structure of R 2l ), and
F(θ, I, z) = O(|I| 2 , ||z|| 3 ). In those coordinates, the invariant torus is simply given by
I = 0, z = 0, and this generalizes both the case of an elliptic fixed point (where the
directions (θ, I) are absent) and of a Lagrangian invariant torus (where the directions z
are absent). If the spectrum {±iα1, . . ., ±iαl} of J2lB is simple, one can assume further
that
H(θ, I, z) = ω.I + α. Ĩ + F(θ, I, z),
where Ĩ are the "formal actions" associated to the z variables. Therefore, after some
appropriate scalings we can consider
H(θ, I, z) = ω.I + α. Ĩ + f(θ, I, z)
(D)
H ∈ As, |f|s < ρ
where As is the space of holomorphic functions on the domain
Ds = {(θ, I, z) ∈ (C k /Z k ) × C k × C 2l | |I(θ)| < s, |I| < s, ||z|| < s}.
Under a suitable Diophantine condition on the vector (ω, α) ∈ Rk+l , one can define
polynomials hm and a formal series h∞ depending on J = (I, Ĩ). Birkhoff’s exponential
estimates in this more difficult situation have been obtained in [JV97]. Regarding
Nekhoroshev’s estimates for a generic integrable Hamiltonian which depends both on
actions and formal actions, they can be easily obtained by obvious modifications of our
method. Therefore we can state the following result.
Theorem 4.5. Suppose H is as in (D). Then under a generic condition on h∞,
there exist positive constants a, a ′ , c1, c2 and ρ0 such that for ρ ≤ ρ0, every solution
(θ(t), I(t), z(t)) of H with |J(0)| < 1 satisfies
|J(t) − J(0)| < c1ρ, |t| < exp
ρ −a′
exp(c2a ′ ρ −a
) .
Once again, the condition on h∞ and the values of the exponents are the same.
4.3.0.3. Let us add that one could easily give similar estimates in the discrete case,
that is for exact symplectic diffeomorphisms near an elliptic fixed point, an invariant

110 Generic super-exponential stability for invariant tori
Lagrangian torus or an invariant linearly stable isotropic reducible torus. Even if one has
the possibility to re-write the proof in these settings, the easiest way is to use suspension
arguments, as it is done qualitatively in [Dou88] or quantitatively in [KP94] (see also
[PT97] for a different approach) and deduce stability results in the discrete case from
the corresponding results in the continuous case.
To conclude, let us mention that important examples of invariant tori satisfying
our assumptions (linearly stable, reducible, isotropic) are those given by KAM theory.
However, the latter not only gives individual tori but a whole Cantor family (see [Pös01]
or [AKN06]). In this context, Popov has proved exponential stability estimates for the
family of Lagrangian KAM tori, if the Hamiltonian is analytic or Gevrey ([Pop00] and
[Pop04]). His proof relies on a KAM theorem with Gevrey smoothness on the parameters
(in the sense of Whitney) and some kind of simultaneous Birkhoff normal form over the
Cantor set of tori. We believe that our method should be useful in trying to extend
those results to obtain super-exponential stability under generic conditions. But clearly
this is a more difficult problem, and the first step is to obtain Nekhoroshev’s estimates
in Gevrey regularity for a generic integrable Hamiltonian, the quasi-convex case having
been settled in [MS02].
4.A Generic assumptions
In this appendix, we will show that our assumption (G) is generic, in the sense that it
defines a prevalent set in the infinite dimensional space of formal power series.
4.A.0.1. Let us first recall the definition of Simultaneous Diophantine Morse functions
(SDM in the following). Let G(n, k) be the set of all vector subspaces of R n of dimension
k. We endow R n with the Euclidean scalar product, and given an integer L ∈ N ∗ ,
we define G L (n, k) as the subset of G(n, k) consisting of subspaces whose orthogonal
complement can be generated by integer vectors with components bounded by L. In
the sequel, B will be an arbitrary open ball of R n .
Definition 4.6. A smooth function h : B → R is said to be SDM if there exist γ ′ > 0
and τ ′ ≥ 0 such that for any L ∈ N ∗ , any k ∈ {1, . . .,n} and any Λ ∈ G L (n, k), there
exists (e1, . . .,ek) (resp. (f1, . . .,fn−k)), an orthonormal basis of Λ (resp. of Λ ⊥ ), such
that the function hΛ defined on B by
hΛ(u, v) = h (u1e1 + · · · + ukek + v1f1 + · · · + vn−kfn−k)
satisfies the following: for any (u, v) ∈ B,
for any η ∈ R k \ {0}.
∂uhΛ(u, v) ≤ γ ′ −τ ′
L =⇒ ∂uuhΛ(u, v).η > γ ′ −τ ′
L η,
This definition is inspired by the steepness condition of Nekhoroshev and the quantitative
Morse-Sard theory of Yomdin (see the first chapter for more explanations). It
depends on a choice of coordinates adapted to the orthogonal decomposition Λ ⊕ Λ ⊥ ,
so for Λ ∈ G L (n, k) and (u, v) ∈ B, ∂uhΛ(u, v) is a vector in R k and ∂uuhΛ(u, v) is a
symmetric matrix of size k with real entries.

4.A - Generic assumptions 111
Remark 4.7. Note also that the definition can be stated as the following alternative:
for any (u, v) ∈ B, either we have ∂uhΛ(u, v) > γL −τ or ∂uuhΛ(u, v).η > γL −τ η
for any η ∈ R k \ {0}. Hence for a given function it is sufficient to verify that
∂uuhΛ(u, v).η > γL −τ η for any η ∈ R k \ {0}, and we will use this fact later (in
Theorem 4.11).
4.A.0.2. The set of SDM functions on B with respect to γ ′ > 0 and τ ′ ≥ 0 will be
τ ′
denoted by SDMγ ′ (B), and we will also use the notation
τ ′
SDM (B) =
γ ′ >0
SDM
τ ′
γ
′ (B).
The following theorem was proved in the first chapter, and it relies on non trivial results
from quantitative Morse-Sard theory ([Yom83],[YC04]).
Proposition 4.3. Let τ > 2(n2 + 1), and h ∈ C2n+2 (B). Then for Lebesgue almost all
τ ′
ξ ∈ R n , the function hξ, defined by hξ(I) = h(I) −ξ.I for I ∈ B, belongs to SDM
Now let us recall the definition of a prevalent set ([HSY92], see also [KH08] and
[OY05]).
Definition 4.8. Let E be a completely metrizable topological vector space. A Borel
subset S ⊆ E is said to be shy if there exists a Borel measure µ on E, with 0 < µ(C) < ∞
for some compact set C ⊆ E, such that µ(x + S) = 0 for all x ∈ E.
An arbitrary set is called shy if it is contained in a shy Borel subset, and finally the
complement of a shy set is called prevalent.
For a finite dimensional vector space E, by an easy application of Fubini theorem,
prevalence is equivalent to full Lebesgue measure. The following "genericity" properties
can be checked ([OY05]): a prevalent set is dense, a set containing a prevalent set is also
prevalent, and prevalent sets are stable under translation and countable intersection.
Furthermore, we have an easy but useful criterion for a set to be prevalent.
Proposition 4.4 ([HSY92]). Let A be a Borel subset of E. Suppose there exists a finite
dimensional subspace F of E such that, denoting λF the Lebesgue measure supported on
F, the set x + A has full λF-measure for all x ∈ E. Then A is prevalent.
τ ′
It is an obvious consequence of Proposition 4.3 and Proposition 4.4 that SDM (B)
is prevalent in C2n+2 (B) for τ ′ > 2(n2 + 1).
4.A.0.3. Now let P∞ = R[[X1, . . .,Xn]] be the space of all formal power series in n
variables with real coefficients. It is naturally a Fréchet space, as the projective limit of
the finite dimensional spaces Pm consisting of polynomials in n variables of degree less
than or equal to m. We define the subset
τ ′
S∞ = {h∞
τ ′
∈ P∞ | hm ∈ SDM (B), ∀m ≥ 2},
where hm = m
k=1 hk if h∞ =
k≥1 hk , and we identify the polynomial hm with the
associated function defined on B. Let us also define
D τ ∞ = {h∞ ∈ P∞ | h1(X) = α.X, α ∈ D τ },
(B).

112 Generic super-exponential stability for invariant tori
where D τ is the set of Diophantine vectors of R n with exponent τ, and finally
τ,τ ′
G∞ = Dτ ′
∞ ∩ Sτ ∞ .
τ,τ ′
The set G∞ is the set of formal power series for which condition (G) holds.
Theorem 4.9. For τ > n − 1 and τ ′ > 2(n2 τ,τ ′
+ 1), the set G∞ is prevalent in P∞.
Proof. As the intersection of two prevalent sets is prevalent, it is enough to prove that
both sets Dτ ′
∞ , for τ > n − 1, and Sτ ∞ , for τ ′ > 2(n2 + 1), are prevalent.
For the set Dτ ∞, this is an easy consequence of the fact that Dτ is of full Lebesgue
measure in Rn , for τ > n − 1, and Proposition 4.4 with F = P1, the space of linear
τ ′
forms. For the set S , first note that we can write
∞
where, for an integer m ≥ 2,
τ ′
S∞ =
m≥2
τ ′
S∞,m, τ ′
S∞,m = {h∞
τ ′
∈ P∞ | hm ∈ SDM (B)}.
As a countable intersection of prevalent sets is prevalent, it is enough to prove that for
τ ′
each m ≥ 2, the set S∞,m is prevalent in P∞. But once again this is just a consequence
of Proposition 4.3 and Proposition 4.4 with F = P1 the space of linear forms.
by
For m ≥ 2, the set of polynomials hm for which condition (Gm) is satisfied is given
τ ′
τ ′
Sm = {hm ∈ Pm | hm ∈ SDM (B)},
and the proof of the above theorem immediately gives the following result.
Theorem 4.10. For τ ′ > 2(n2 τ ′
+ 1), the set Sm is of full Lebesgue measure in Pm.
4.A.0.4. Now in the special case m = 2, we can state a refined result which is due to
Niederman ([Nie07a]).
Theorem 4.11. For Lebesgue almost all β ∈ Sn(R), the function
h(I) = α.I + βI.I
τ ′
belongs to SDM (B) provided τ ′ > n2 + 1.
In the above theorem, there is no condition on α, and contrary to Proposition 4.3,
the proof does not rely on Morse-Sard theory. Let us denote by λ the one-dimensional
Lebesgue measure and by Ik the identity matrix of size k. We shall use the following
elementary lemma.
Lemma 4.12. Let k ∈ {1, . . ., n}, βk ∈ Sk(R) and κ > 0. Then there exists a subset
Cκ ⊆ R such that
λ(Cκ) ≤ 2kκ,
and for any ξ /∈ Cκ, the matrix βk,ξ = βk − ξIk satisfies
for any η ∈ R k \ {0}.
βk,ξ.η > κη,

4.A - Generic assumptions 113
Of course, the set Cκ depends on the matrix βk.
Proof. Let {λ1, . . .,λk} be the eigenvalues of βk, then in an orthonormal basis of eigenvectors
for βk, the matrix βk,ξ is also diagonal, with eigenvalues {λ1 − ξ, . . ., λk − ξ}.
Then one has βk,ξ.η > κη for any η ∈ R k \ {0} provided that for all i ∈ {1, . . .,k},
|λi − ξ| > κ, that is if ξ does not belong to
Cκ =
k
[λi − κ, λi + κ].
i=1
The measure estimate λ(Cκ) ≤ 2kκ is trivial.
With this lemma, the proof is now similar to that of Proposition 4.3.
Proof of Theorem 4.11. Let h(I) = α.I + βI.I, and given Λ ∈ GL (n, k), we denote
by βΛ ∈ Sk(R) the matrix which represents the quadratic form βI.I restricted to the
subspace Λ. Since the second derivative of h along any subspace is constant, then coming
τ ′
back to definition 4.6 and using remark 4.7, h ∈ SDM ′ if
γ
βΛ.η > γ ′ −τ ′
L η, (36)
for any Λ ∈ GL (n, k) and any η ∈ Rk τ ′
\ {0}. Let Aγ ′ be the subset of Sn(R) whose
elements contradict condition (36), that is
and
τ ′
Aγ ′ = {β ∈ Sn(R) | βΛ.η ≤ γ ′ τ ′
L η, Λ ∈ G L (n, k), η ∈ R k \ {0}},
τ ′
What we need to show is that A
n2 + 1.
τ ′
A =
γ ′ >0
τ ′
Aγ ′.
has zero Lebesgue measure in Sn(R) provided τ ′ >
Apply Lemma 4.12 to βΛ ∈ Sk(R), with κ = γ ′ −τ ′
L , to have a subset Cγ ′ ,τ ′ ,L,Λ ⊆ R
such that
λ(Cγ ′ ,τ ′ ,L,Λ) ≤ 2kγ ′ −τ ′
L , (37)
and for any ξ /∈ Cγ ′ ,τ ′ ,L,Λ, the matrix βΛ,ξ = βΛ − ξIk satisfies
for any η ∈ R k \ {0}. If we define
then
and so
Cγ ′
,τ ′ =
βΛ,ξ.η > γ ′ −τ ′
L η
L∈N∗ k∈{1,...,n} Λ∈GL (n,k)
Cγ ′ ,τ ′ ,L,Λ,
Cγ ′ ,τ ′ = {ξ ∈ R | βξ
τ ′
= β − ξIn ∈ Aγ ′}
Cτ ′ =
γ ′ >0
Cγ ′ ,τ ′ = {ξ ∈ R | βξ
τ ′
∈ A }.

114 Generic super-exponential stability for invariant tori
It remains to prove that the Lebesgue measure of Cτ ′ is zero, since by Fubini theorem,
τ ′
this will imply that the Lebesgue measure of A is zero. By our estimate (37), we have
λ(Cγ ′
,τ ′) ≤
n
L∈N∗ k=1
≤
n
L∈N∗ k=1
|G L (n, k)|2kγ ′ −τ ′
L
L n2
2kγ ′ −τ ′
L
n
= 2 k L n2−τ ′
γ ′
k=1
L∈N ∗
= n(n + 1) L n2−τ ′
γ ′
L∈N ∗
and, since τ ′ > n 2 + 1, the above series is convergent. Hence
λ(Cτ ′) = inf
γ ′ >0 λ(Cγ ′ ,τ ′) = 0.

5.1 - Introduction 115
5 Polynomial stability for C k quasi-convex Hamiltonian
systems
A major result about perturbations of integrable Hamiltonian systems is the Nekhoroshev
theorem, which gives exponential stability for all solutions provided the system is
analytic and the integrable Hamiltonian is generic. In the particular but important case
where the latter is quasi-convex, these exponential estimates have been generalized by
Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by
Lochak in the analytic case. In this chapter, using the same approach, we investigate
the situation where the Hamiltonian is assumed to be only finitely differentiable, for
which it is known that exponential stability does not hold but nevertheless we prove
estimates of polynomial stability. The results of this chapter are contained in [Bou10b].
5.1 Introduction
In this chapter, we are concerned with the stability properties of near-integrable Hamiltonian
systems of the form
H(θ, I) = h(I) + f(θ, I)
|f| < ε 3, the famous example of Arnold ([Arn64]) shows that there exist
"unstable" solutions, along which the variation of the actions can be arbitrarily large no
matter how small the perturbation is. From its very beginning, KAM theory was known
to hold for non-analytic Hamiltonians (see [Mos62] in the context of twist maps). It
is now well established in various regularity classes, including the C ∞ case (essentially
by Herman, see [Bos86] and [Féj04]) and the Gevrey case ([Pop04]). Following ideas
of Moser, the theorem also holds if H is only of class C k , with k > 2n (see [Pös82],
[Sal04], [SZ89] and also [Alb07] for a refinement), even though the minimal number of
derivatives is still an open question, except in a special case for n = 2 ([Her86], see also
[KT09] for the related context of linearization of circle diffeomorphisms).
5.1.0.2. Another fundamental result, which complements KAM theory, is given by
Nekhoroshev’s theorem ([Nek77], [Nek79]). If the integrable part h satisfies some generic

116 Polynomial stability for C k quasi-convex Hamiltonian systems
condition and the system is analytic, then for ε sufficiently small there exist positive
constants c1, c2, c3, a and b such that
|I(t) − I0| ≤ c1ε b , |t| ≤ c2 exp(c3ε −a ),
for all initial actions I0. Hence all solutions are stable, not for all time, but for an
exponentially long time. In the special case where h is strictly quasi-convex, a completely
new proof of these estimates was given by Lochak ([Loc92]) using periodic averaging
and simultaneous Diophantine approximation. The method of Lochak has had many
applications, in particular it was used by Marco and Sauzin to extend Nekhoroshev’s
theorem to the Gevrey regular case under the quasi-convexity assumption ([MS02]).
5.1.0.3. However, no such estimates have been studied when the Hamiltonian is merely
finitely differentiable, and this is the content of the present chapter. We will prove below
(Theorem 5.1) that if H is of class Ck , for k ≥ 2, and h quasi-convex, then one has the
stability estimates
|I(t) − I0| ≤ c1ε 1
k−2
2n,
−
|t| ≤ c2ε 2n ,
for some positive constants c1 and c2, and provided that ε is small enough. Of course,
under our regularity assumption the exponential estimates have been replaced with polynomial
estimates, and earlier examples show that exponential stability cannot possibly
hold under such a weak regularity assumption (this is discussed in [MS04]). The proof
will use once again the ideas of Lochak which, among other things, reduces the analytic
part to its minimum and we will also follow the implementation of Marco and Sauzin
in the Gevrey case.
5.1.0.4. As we recalled above, KAM theory for finitely differentiable Hamiltonian
systems has been widely studied, and so we believe that Nekhoroshev’s estimates under
weaker regularity assumptions have their own interest. Moreover, for obvious reasons,
examples of unstable solutions (so-called Arnold diffusion) are more easily constructed
in the non-analytic case, and it is a natural question to estimate the time of instability
(see [KL08a] and [KL08b] for examples of class C k with a polynomial time of diffusion).
Finally, one of our motivations is to generalize these estimates using the method of the
first chapter, where Lochak’s ideas are extended to deal with analytic but C k -generic
unperturbed Hamiltonians, with k > 2n + 2.
5.2 Main result
5.2.0.1. Let T n = R n /Z n , and consider a Hamiltonian function H defined on the
domain
DR = T n × BR,
where BR is the open ball of R n around the origin of radius R, with respect to the
supremum norm | . |. As usual, we shall occasionally identify H with a function defined
on R n × BR which is 1-periodic with respect to the first n variables.
We assume that H is of class C k , for an integer k ≥ 2, i.e. it is k-times differentiable
and all its derivatives up to order k extend continuously to the closure DR. We denote

5.2 - Main result 117
by Ck (DR) the space of such functions, which is a Banach space with the norm
|H| Ck (DR) = sup sup
0≤l≤k |α|=l
sup
x∈DR
|∂ α H(x)|
where x = (θ, I), α = (α1, . . ., α2n) ∈ N 2n , |α| = α1 + · · · + α2n and
∂ α = ∂ α1
1 . . .∂ α2n
2n .
In the case where the Hamiltonian H = h depends only on the action variables, we will
simply write |h| C k (BR).
Our Hamiltonian H ∈ Ck (DR) is assumed to be Ck-close to integrable, that is, of
the form
H(θ, I) = h(I) + f(θ, I)
|f| Ck (DR) < ε

118 Polynomial stability for C k quasi-convex Hamiltonian systems
know what the optimal exponents should be. However, using the geometric arguments
of the next chapter, one can easily improve the stability exponent a in order to obtain
for δ > 0 but arbitrarily small.
a =
k − 2
− δ,
2(n − 1)
Let us finally point out that if H is C ∞ , then it is an immediate consequence of the
above result that the action variables are stable for an interval of time which is longer
than any prescribed power of ε −1 , but even in this case exponential stability does not
hold.
5.2.0.3. As in the analytic or Gevrey case, we can also state a refined result near
resonances. Suppose Λ is a submodule of Z n of rank m, d = n − m and let SΛ be the
corresponding resonant manifold, that is
SΛ = {I ∈ BR | k.∇h(I) = 0, k ∈ Λ}.
We can prove the following statement, which actually contains the previous one.
Theorem 5.2. Under the previous hypotheses, assume d(I(0), SΛ) ≤ σ √ ε for some
constant σ > 0, and set
k − 2
ad =
2d , bd = 1
2d .
Then there exist ε ′ 0 , c′ 1 and c′ 2 such that if ε ≤ ε′ 0
, one has
|I(t) − I(0)| ≤ c ′ 1 εbd , |t| ≤ c ′ 2 ε −ad .
For Λ = {0}, d = n and SΛ = BR/2, we recover Theorem 5.1 and therefore it will be
enough to prove Theorem 5.2.
5.2.0.4. The constants ε0, c1 and c2 depend only on h, more precisely they depend on
k, n, R, M and m while the constants ε ′ 0 , c′ 1 and c′ 2 also depend on σ and Λ. However we
will not give explicit values for them in order to avoid complicated and rather useless
expressions. Hence we shall replace them by the symbol · when it is convenient: for
instance, we shall write u

5.3 - Analytical part 119
is given by those harmonics associated with integers k ∈ Z n in resonance with ω, that
is such that k.ω = 0. Actually one can construct a symplectic, close-to-identity transformation
Φ defined around I, such that
H ◦ Φ = h + g + ˜ f
where g contains only harmonics in resonance with ω and ˜ f is a small remainder. These
are usually called resonant normal forms, and to obtain them one has to deal with
small divisors k.ω which involve technical estimates. If the system is analytic, the above
remainder ˜ f can be made exponentially small with respect to the inverse of the size of the
perturbation, as was first shown by Nekhoroshev. But for finitely differentiable systems
one might guess that the remainder can only be polynomially small, even though this
should be difficult (or at least technical) to prove using the usual approach.
5.3.0.2. It is a remarkable fact discovered by Lochak ([Loc92]) that to prove exponential
estimates in the quasi-convex case with the analyticity assumption, it is enough to
average along periodic frequencies, which are frequencies ω such that Tω ∈ Z n \ {0}
for some T > 0 (see also the first chapter for an extension of this method for generic
integrable Hamiltonians). These periodic frequencies correspond to periodic orbits of
the unperturbed Hamiltonian, hence in this approach no small divisors arise. As a
consequence this special resonant normal form is much easier to obtain. The aim of this
section is to construct such a normal form, up to a polynomial remainder. This will be
done in 5.3.3. But first we will recall some useful estimates concerning the C k norm
in 5.3.1, and then prove an intermediate statement in 5.3.2.
5.3.1 Elementary estimates
5.3.1.1. Let us begin by recalling some easy estimates. Given two functions f, g ∈
C k (DR), the product fg belongs to C k (DR) and by the Leibniz rule
|fg| C k (DR)

120 Polynomial stability for C k quasi-convex Hamiltonian systems
where
∂If = (∂I1f, . . . , ∂Inf), ∂θf = (∂θ1f, . . . , ∂θnf).
Obviously Xf ∈ C k−1 (DR, R 2n ), and trivially
|Xf| C k−1 (DR) ≤ |f| C k (DR).
Moreover, by classical theorems on ordinary differential equations, if Xf is of class
C k−1 then so is the time-t map Φ f
t of the vector field Xf, when it exists. Assuming
|∂θf|C 0 (DR) < r for some r < R (for example |Xf|C 0 (DR) < r), then by the mean value
theorem
Φ f = Φ f
1 : DR−r −→ DR
is a well-defined C k−1 -embedding. In the case where f is integrable, one can choose
r = 0.
In the sequel, we will need to estimate the C k norm of Φ f in terms of the C k norm of
the vector field Xf. More precisely we need the rather natural fact that Φ f is C k -close
to the identity when Xf is C k -close to zero. This is trivial for k = 0. In the general
case, this follows by induction on k using on the one hand the relation
Φ f
t = Id +
t
0
Xf ◦ Φ f sds,
and on the other the formula of Faà di Bruno (see [AR67] for example), which gives
bounds of the form
and also
|F ◦ G| C k

5.3 - Analytical part 121
5.3.2 The linear case
Following [MS02], we change for a moment our setting and we consider a perturbation
of a linear Hamiltonian, more precisely the Hamiltonian
H(θ, I) = l(I) + f(θ, I)
(∗∗)
|f| Ck (Dρ) < µ 0 is fixed and l(I) = ω.I is a linear Hamiltonian with a T-periodic frequency
ω. Recall that this means that
T = inf{t > 0 | tω ∈ Z n \ {0}}
is well-defined. In this context, our small parameter is µ.
In the proposition below, we will construct a "global" normal form for the Hamiltonian
(∗∗), which we will use in the next section to produce a "local" normal form
around periodic orbits for our original Hamiltonian (∗).
Proposition 5.1. Consider H as in (∗∗) with k ≥ 2, and assume
Then there exists a C 2 symplectic transformation
with |Φ − Id| C 2 (Dρ/2)

122 Polynomial stability for C k quasi-convex Hamiltonian systems
and for j ∈ {0, . . ., k − 2}, let
ρj = ρ − jr ≥ ρ/2.
Then we claim that for any j ∈ {0, . . ., k − 2}, there exists a C k−j symplectic transformation
Φj : Dρj → Dρ with |Φj − Id| C k−j (Dρj )

5.3 - Analytical part 123
Indeed, thanks to the condition Tµ

124 Polynomial stability for C k quasi-convex Hamiltonian systems
5.3.3 Normal form
Now let us come back to our original setting which is the Hamiltonian
H(θ, I) = h(I) + f(θ, I)
|f| C k (DR) < ε 0 be such that
Then there exists a C 2 symplectic transformation
with |ΠIΦ − IdI| C 0 (B(I∗,µ))

5.3 - Analytical part 125
which sends the domain D2 = T n × B2 onto T n × B(I∗, 2µ), and note that by the
condition µ

126 Polynomial stability for C k quasi-convex Hamiltonian systems
Observe that (µl + µ 2 hµ) ◦ σ −1
µ = h, so we may set
and write
It is obvious that {g, l} = 0 with
and similarly
so
Moreover, as ∂˜ θ ˜ f = µ∂˜ θ ˜ fµ then
and finally
is trivial. This ends the proof.
5.4 Proof of Theorem 5.2
g = µ ˆ fµ ◦ σ −1
µ , ˜ f = µ ˜ fµ ◦ σ −1
µ ,
H ◦ Φ = h + g + ˜ f.
|g| C 0 (T n ×B(I∗,µ)) ≤ µ| ˆ fµ| C 0 (D1)

5.4 - Proof of Theorem 5.2 127
and the period T satisfies
1 0 and
˜H = h + g + ˜ f ∈ C 2 (T n × B(I∗, r))
with h satisfying (C), {g, l} = 0 and the estimates
If
|g + ˜ f|C 0 (T n ×B(I∗,r)) < r 2 , |∂˜ θ ˜ f|C 0 (T n ×B(I∗,r)) < r 2 τ −1 .
r

128 Polynomial stability for C k quasi-convex Hamiltonian systems
and the period T trivially satisfies T 1, we apply Proposition 5.3
with
d−1
−
Q =· ε 2d .
1
−
and the condition (43) gives a first smallness condition on ε. Observe that Q d−1 =· ε 1
2d,
hence the periodic action I∗ given by the proposition satisfies
|I0 − I∗|

Part III
From stability to instability
Summary
6 Improved exponential stability for quasi-convex Hamiltonian systems
131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 The analytic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 The Gevrey case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
129

130

6.1 - Introduction 131
6 Improved exponential stability for quasi-convex
Hamiltonian systems
In this chapter, we improve previous results on exponential stability for analytic and
Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular,
this provides a sharper lower bound on the time of Arnold diffusion which we believe to
be optimal. This chapter is based on [BM10].
6.1 Introduction
6.1.0.1. This chapter deals with some stability properties of near-integrable Hamiltonian
systems of the form
H(θ, I) = h(I) + f(θ, I)
|f| < ε 0, no matter how small the perturbation is. Such instability
is commonly referred to as Arnold diffusion. Obviously Nekhoroshev’s estimates give a
lower bound on the diffusion time τ(ε) (or equivalently an upper bound on the diffusion
speed) which is exponentially large (or exponentially small when referring to the rate of
diffusion).
6.1.0.2. This chapter is concerned with the precise time scale at which stability breaks
down and instability takes place, by which we mean the precise value of the exponent

132 Improved exponential stability for quasi-convex Hamiltonian systems
a, in the special case where the unperturbed Hamiltonian h is quasi-convex (that is, its
energy sub-levels are convex).
The quasi-convex case, which is of both practical and theoretical interest, has been
widely studied in Nekhoroshev theory, essentially for two reasons. First, the proof
is much easier in this situation and a more refined result is available: the stability
exponents may be chosen as
a = b = (2n) −1 .
These facts are best illustrated by the striking proof given by Lochak ([Loc92], see also
[LN92] and [LNN94]), though they are also accessible via a more traditional approach
as was shown by Pöschel ([Pös93]). Note also that these values have been generalized by
Niederman in the steep case ([Nie04]) using both ideas of Lochak and Pöschel. Yet there
is another reason for which the quasi-convex case is interesting, which is the so-called
stabilization by resonances: if a solution starts close to a resonance of multiplicity m,
m < n, then it possesses better stability properties, described by the "local" exponents
am = bm = (2(n − m)) −1 .
This is a quite surprising fact, as it shows that even though resonances are the main
cause of diffusion, at the same time they improve finite time stability. However, this
property certainly does not hold without some convexity assumption (in the steep case
for instance).
6.1.0.3. The optimality of the exponent a, in connection with the minimal time of
instability, was first studied by Bessi who introduced powerful variational methods to
revisit Arnold’s example and estimate the time of diffusion. In [Bes96] and [Bes97b],
he proved that the latter is of order exp ε−1
− 2 for n = 3 and exp ε 1
4 for n = 4.
Moreover, in Bessi’s example the solution passes close to a double resonance, and so
the time is the best possible in this case, in view of the values of the local exponent
a2 for n = 3 and n = 4. Recently, using similar variational arguments, this result has
been generalized to an arbitrary number of degrees of freedom n by Ke Zhang ([Zha09]),
namely he constructed a special orbit passing close to a double resonance for which the
time of diffusion is estimated by exp 1
−
ε 2(n−2) .
Another approach has been proposed by Marco-Sauzin ([MS02]) and Lochak-Marco
([LM05]), following novel ideas of Herman. In [MS02] the authors show that Nekhoroshev’s
estimates extend to perturbations of quasi-convex Hamiltonians which are α-
Gevrey regular, α ≥ 1, with the exponents
and local exponents
a = (2αn) −1 , b = (2n) −1
am = (2α(n − m)) −1 , bm = (2(n − m)) −1 .
Note that 1-Gevrey functions are exactly analytic functions, and basically when α ranges
from one to infinity α-Gevrey functions interpolate between analytic and C∞ functions.
Therefore this result generalizes the estimates in the analytic case. Using a geometric
mechanism different and more precise than Arnold’s, in [MS02] the authors constructed
a drifting orbit with a time of order exp ε −
1
2α(n−2) in the non-analytic case, that is when

6.2 - Main results 133
α > 1. Adding some more technical ideas, it is shown in [LM05] that the example also
works in the analytic case, but the time is estimated as exp 1
−
ε 2(n−3) , which is only close
to optimal (however refinements are certainly possible to reach the value (2(n − 2)) −1
in this class of examples).
Therefore, if the unperturbed Hamiltonian is quasi-convex, the best exponent of
stability a up to now satisfies
in the analytic case and more generally
(2n) −1 ≤ a < (2(n − 2)) −1
(2αn) −1 ≤ a < (2α(n − 2)) −1
in the Gevrey case. The goal of this chapter is to improve the lower bound both in the
analytic or Gevrey case, so as to have
and
(2(n − 1)) −1 − δ ≤ a < (2(n − 2)) −1
(2α(n − 1)) −1 − δ ≤ a < (2α(n − 2)) −1
for δ > 0 but arbitrarily small (see Theorem 6.1 and Theorem 6.2 in the next section).
We believe that this bound is optimal, in the sense that one could reach the value
(2(n − 1)) in Arnold diffusion, using a significantly different mechanism of instability.
6.2 Main results
6.2.0.1. In order to state our main results, let us now describe our setting more
precisely, beginning with the analytic case. Let B = B(0, R) be the open ball of R n of
radius R > 0, with respect to the supremum norm | . |, centered at the origin. Given
s > 0, we let As(D) be the space of bounded real-analytic functions on D = T n × B
which extend as holomorphic functions on the complex domain
Ds = {(θ, I) ∈ (C n /Z n ) × C n | |I(θ)| < s, d(I, B) < s},
and which are continuous on the closure of Ds. Here we denoted by I(θ) the imaginary
part of θ, by | . | the supremum norm on C n , and by d the associated distance on C n . It
is well-known that As(D) is a Banach space with its usual supremum norm | . |s, where
In the following, we shall denote by
|f|s = sup |f(z)|, f ∈ As(D).
z∈Ds
Bs = {I ∈ R n | d(I, B) < s}
the real part of our domain Ds in action space. The geometric parameters n, R, s are
assumed to be chosen once and for all in the following.

134 Improved exponential stability for quasi-convex Hamiltonian systems
We now introduce the parameters related to the choice of the system. The integrable
part h : Bs → R will be assumed to be strictly quasi-convex: the gradient map ∇h does
not vanish and there exists a positive number m such that
∇ 2 h(I)v.v ≥ m|v| 2
(QC(m))
holds for any I ∈ Bs and any v orthogonal to ∇h(I) (with respect to the Euclidean
scalar product). Moreover, the derivatives up to order 3 of h on Bs are assumed to be
bounded: there exists M > 0 such that for all I ∈ Bs, one has
|∂ k h(I)| ≤ M, 1 ≤ |k1| + · · · + |kn| ≤ 3. (B(M))
Therefore we will consider systems of the form
⎧
H(θ, I) = h(I) + f(θ, I),
⎪⎨
h ∈ As(D), f ∈ As(D),
⎪⎩
h satisfies (QC(m)) and (B(M)),
|f|s < ε.
(C(M, m, ε))
Note that we suppress of the geometric parameters in the notation. In the following
we will call a stable constant (in the analytic case) any positive constant c which depends
on the whole set of parameters, that is n, R, s, M, m, together with a parameter δ or
ρ to be defined below, but not on a particular choice of H satisfying the condition
(C(M, m, ε)).
6.2.0.2. The main result of the chapter is the following.
Theorem 6.1. Consider a real number δ satisfying
0 < δ ≤ (2n(n − 1)) −1 .
Then there exist stable constants c1, c2, c3 and ε0 such that if 0 ≤ ε ≤ ε0, and if H
satisfies (C(M, m, ε)), the following estimates
|I(t) − I0| ≤ c1ε δ(n−1) 1
−
, |t| ≤ c2 exp c3ε 2(n−1) +δ
hold true for every initial action I0 ∈ B(0, R/2).
Moreover, consider a real number ρ satisfying 0 < ρ < R/2. Then there exist stable
constants c4, c5 and ˜ε0 such that if 0 ≤ ε ≤ ˜ε0, then
for every I0 ∈ B(0, R/2).
|I(t) − I0| ≤ ρ, |t| ≤ c4 exp 1
−
c5ε 2(n−1)
Choosing our constant δ arbitrarily close to zero, our result ensures stability for a
time scale which is arbitrarily close to exp 1
−
ε 2(n−1) , therefore we improve the previous
results of stability obtained independently by Lochak-Neishtadt ([Loc92] and [LN92])
and Pöschel ([Pös93]), which were believed to be optimal.

6.2 - Main results 135
In fact in the extreme case where δ = (2n(n −1)) −1 , which in our situation gives the
worst stability time (but of course the best radius of confinement), our result reads
|I(t) − I0| ≤ c1ε 1
2n, |t| ≤ c2 exp 1
−
c3ε 2n
and we recover the previous result of stability. Hence, when our parameter δ ranges from
(2n(n −1)) −1 to zero, our theorem "interpolates" between previous stability results and
what should be the optimal stability.
Indeed, in the other extreme case which corresponds to the second part of our theorem,
our result does not give stability, since the radius of confinement can be arbitrarily
small but no longer tends to 0 with ε. We believe that this is not an artefact of the
method and that instability should occur at this precise time scale, at a time of order
exp 1
−
ε 2(n−1) . We plan to construct an example with an unstable orbit which has a drift
of order one during such a time interval. This necessitates the use of a more refined
instability mechanism in the neighbourhood of double resonances, a topic which is also
crucial in connection with the problem of genericity of Arnold diffusion.
6.2.0.3. Our result also holds if the Hamiltonian is only Gevrey regular. Let us recall
that given α ≥ 1 and L > 0, a function H ∈ C∞ (D) is (α, L)-Gevrey if, using the
standard multi-index notation, we have
|H|α,L =
l∈N 2n
L |l|α (l!) −α |∂ l H|D < ∞
where | . |D is the usual supremum norm for functions on D. The space of such functions,
with the above norm, is a Banach space that we denote by G α,L (D). Analytic functions
are a particular case of Gevrey functions, as one can check that G 1,L (D) = AL(D).
Let us introduce the main condition on the Hamiltonian systems in the Gevrey case
⎧
H(θ, I) = h(I) + f(θ, I),
⎪⎨
h ∈ G
⎪⎩
α,L (D), f ∈ Gα,L (D),
(C(α, L, M, m, ε))
h satisfies (QC(m)) and (B(M)),
|f|α,L < ε.
We now call a stable constant (in the Gevrey case) any positive constant c which
depends on the whole set of parameters, that is α, L, n, R, M, m, together with a parameter
δ or ρ which will be defined below, but not on a particular choice of H satisfying
the condition (C(α, L, M, m, ε)).
6.2.0.4. Our second result is the following Gevrey version of Theorem 6.1.
Theorem 6.2. Consider a real number δ satisfying
0 < δ ≤ (2αn(n − 1)) −1 .
Then there exist stable constants c ′ 1, c ′ 2, c ′ 3 and ε ′ 0 such that if 0 ≤ ε ≤ ε ′ 0, and if H
satisfies C(α, L, M, m, ε), the following estimates
|I(t) − I0| ≤ c ′ 2δ
1ε 5(n−1), |t| ≤ c ′ 2 exp
c ′ 3ε− 1
2α(n−1) +δ
,

136 Improved exponential stability for quasi-convex Hamiltonian systems
hold true for every initial action I0 ∈ B(0, R/2).
Moreover, consider a real number ρ satisfying 0 < ρ < R/2. Then there exist stable
constants c ′ 4 , c′ 5 and ˜ε′ 0 such that if 0 ≤ ε ≤ ˜ε′ 0 , then
for every I0 ∈ B(0, R/2).
|I(t) − I0| ≤ ρ, |t| ≤ c ′ 4 exp c ′ 5ε− 1
2α(n−1)
The same remarks as above apply in the Gevrey case. In particular δ can be chosen
arbitrarily close to zero and our result ensures stability for an interval of time which is
arbitrarily close to exp ε −
1
2α(n−1) . However, our radius of stability is worse than in the
analytic case, so we do not fully recover the result obtained in [MS02], but of course the
time of stability is the most important issue.
6.2.0.5. To avoid cumbersome expressions in the following, when there is no risk of
confusion we will replace the stable constants with a dot. More precisely, an assertion
of the form "there exists a stable constant c such that f < c g" will be simply replaced
with "f

6.3 - The analytic case 137
Given a real number K ≥ 1, we will say that Λ is a K-submodule if it admits a Z-basis
{k 1 , . . .,k r } satisfying |k i |1 ≤ K for i ∈ {1, . . ., r}, where
|k i |1 = |k i 1| + · · · + |k i n|.
As usual, it is enough to consider only maximal submodules, which are those that are not
strictly contained in any other submodule of the same rank. Given such a submodule,
we define its volume by
|Λ| = √ det t MM
where M is any n × r matrix whose columns form a basis for Λ (this is easily seen to
be independent of the choice of such a matrix). The following stability theorem is due
to Pöschel.
Theorem 6.3 (Pöschel). Let Λ be a K-submodule of Z n of rank r, with r ∈ {0, . . .,n−
1}. Assume that ε ≥ 0 and K ≥ 1 satisfy
ε|Λ| 2 K 2(n−r)

138 Improved exponential stability for quasi-convex Hamiltonian systems
Lemma 6.4. Let Λ be a K-submodule of Z n of rank 1. Assume that ε ≥ 0 and K ≥ 1
satisfy
εK 2n

6.3 - The analytic case 139
Lemma 6.6. Let I be a closed interval of length l > 0 contained in [−1, 1]. Then there
exists a rational number p/q ∈ I ∩ Q satisfying
|q| + |p| < (4 √ 2)l −1 2.
The exponent in l −1
2 comes from the use of Dirichlet’s theorem on the approximation
of real numbers by rational ones, but this result is not necessary, as in the sequel a trivial
bound of order l −1 would be enough.
Proof. Let us write I = [x − l/2, x + l/2] for some x ∈ [−1, 1], and let q be the smallest
integer larger than √ 2l−1 2, that is
√ 2l − 1
2 ≤ q < √ 2l −1
2 + 1.
By Dirichlet’s theorem there exists an integer p ∈ Z such that
|x − p/q| < q −2 .
Since √ 2l −1
2 ≤ q, we have q −2 ≤ l/2, and so
|x − p/q| < l/2
which means that p/q ∈ I. Moreover, as I ⊂ [−1, 1], then |p| ≤ q and
|q| + |p| ≤ 2q.
Recall that q < √ 2l −1
2 + 1, but since I ⊆ [−1, 1], we have l ≤ 2, so that 1 ≤ √ 2l −1
2 and
therefore
This gives
which concludes the proof.
q < (2 √ 2)l −1 2.
|q| + |p| < (4 √ 2)l −1
2
The following result is our main lemma. It essentially says that a (long) drifting
orbit has to cross a simple resonance, since all other orbits are stable on the interval of
time over which they are defined.
Lemma 6.7. Consider ε ≥ 0 and K ≥ 1 such that
K −2

140 Improved exponential stability for quasi-convex Hamiltonian systems
Once again, the exponent in K −2 comes from Lemma 6.6 and hence from Dirichlet’s
theorem, but a bound of order K −1 would be sufficient for the final result. Let us also
add that if τ is the maximal time of existence of the solution within the initial domain
T n × B, then our stability estimate easily ensures that τ = +∞, which in turn implies
stability for all time (see the proof of Theorem 6.1).
Proof. We will make conditions (47) explicit and prove that
when
|I(t) − I0| < 32 C −1 K −2 , 0 ≤ t < τ,
K −2 < (32) −1 Cρ0, εK 2 < 16,
where ρ0 and C are the stable constants of Lemma 6.5.
We will argue by contradiction, so we assume that there exists a time ˜t for which
Consider the curve
and let ρ = 32 C −1 K −2 . Then
|I(˜t) − I0| ≥ 32 C −1 K −2 .
σ(t) = (I(t), |ω(t)| −1 ) ∈ B × R +
t ∗ = inf{t ∈ [0, τ[ | σ(t) /∈ B(σ(0), ρ)}
is well-defined, as the above set contains ˜t. Now ρ < ρ0, so we can apply Lemma 6.5: the
restriction of Ψh to the open ball B(σ(0), ρ) is a diffeomorphism whose image contains
the closed ball B(Ψh(σ(0)), 32 K −2 ). Considering a slightly larger ball over which Ψh
remains a diffeomorphism, this easily implies that
Ψh(σ(t ∗ )) /∈ B(Ψh(σ(0)), 32 K −2 ),
that is, h(I(t ∗ )), |ω(t ∗ )| −1 ω(t ∗ ) − h(I0), |ω(0)| −1 ω(0) ≥ 32 K −2 .
Using the conservation of energy, one has
so that necessarily
|h(I(t ∗ )) − h(I0)| < 2ε < 32 K −2
|ω(t ∗ )| −1 ω(t ∗ ) − |ω(0)| −1 ω(0) ≥ 32 K −2 .
Therefore there exists an index i ∈ {1, . . ., n} such that
ωi(t
∗ )
|ω(t∗
ωi(0)
−
)| |ω(0)| ≥ 32 K−2
and this estimate means that the image of the interval [0, t∗ ] under the continuous
function
t ↦−→ ωi(t)
∈ [−1, 1]
|ω(t)|

6.3 - The analytic case 141
contains a non trivial interval I of length l = 32 K −2 . Now we can apply Lemma 6.6 to
find a rational number p/q ∈ Q in reduced form and a time t ′ ∈ [0, t ∗ ] such that
with
ωi(t ′ )
|ω(t ′ )|
= p
q
(48)
|p| + |q| < 4 √ 2(32 K −2 ) −1
2 = K. (49)
But since |ω(t ′ )| = |ωj(t ′ )| for some j ∈ {1, . . ., n}, and replacing p with −p if ωj(t ′ ) is
negative, the equality (48) can be written as
qωi(t ′ ) − pωj(t ′ ) = 0. (50)
Now let us write k ′ = qei − pej ∈ Z, then from (49) and (50) we have
k ′ .ω(t ′ ) = 0, |k ′ |1 < K,
and since the submodule generated by k ′ is maximal, since p and q are co-prime, we find
that ω(t ′ ) ∈ RK. This gives the desired contradiction.
6.3.0.4. We finally arrive to the proof of Theorem 6.1.
Proof of Theorem 6.1. We choose K of the form
ε0
K = K0
ε
with suitable stable constants K0 and ε0 so that conditions (46) and (47) are satisfied if
γ
0 ≤ ε ≤ ε0, 0 < γ ≤ (2n) −1 .
With this threshold and these bounds on γ, both Lemma 6.4 and Lemma 6.7 can be
applied. Let (θ0, I0) ∈ T n × B(0, R/2) and T be the maximal time of existence within
T n × B(0, R) of the solution (θ(t), I(t)) starting at (θ0, I0). We have to distinguish two
cases.
In the first case, we assume that ω(t) ∈ NK for all t < T. Then we apply Lemma 6.7
with τ = T to get
|I(t) − I0| 0), is included in
B(0, R). Therefore this solution is defined for all time, that is T = +∞, and so the
previous estimate gives
|I(t) − I0|

142 Improved exponential stability for quasi-convex Hamiltonian systems
Again, taking ε0 small enough we can ensure that I(t ∗ ) ∈ B(0, R/2), then we can apply
Lemma 6.4 to the solution It ∗(t) = I(t + t∗ ), whose initial frequency belongs to RK, to
obtain
Setting
this gives
|It∗(t) − It∗(0)|

6.4 - The Gevrey case 143
6.4 The Gevrey case
In this section, we will prove Theorem 6.2. In fact, it will be enough to have a version of
Lemma 6.4 in the Gevrey case, as the geometric considerations of the previous section
still apply with no changes.
Here we shall use a result from [MS02], which follows the method introduced by
Lochak ([Loc92]) from which the results of improved stability near resonances actually
originate. In the latter approach, the notion of "order" of a resonance is more intrinsic,
however it is also more difficult to compute.
Let Λ be a submodule of Zn of rank r, with r ∈ {1, . . ., n}, and choose a basis
{k1 , . . .,k r } ∈ Zn for Λ. We define the matrix L of size r ×n with integer entries whose
rows are given by the vectors ki = (ki 1 , . . .,ki n ), 1 ≤ i ≤ r, that is
⎛ ⎞
k 1 1 · · · k 1 n
⎜
L = ⎝ . .
kr 1 · · · kr ⎟
⎠ ∈ Mr,n(Z).
n
Then it is an elementary result of linear algebra that there exists integers d1, . . .,dr ∈ Z,
satisfying the divisibility conditions d1| . . . |dr, such that L is equivalent to the diagonal
matrix
⎛
d1
⎜
∆ = ⎝
.. .
0 0 · · ·
.
⎞
0
⎟
⎠ ∈ Mr,n(Z).
0 dr 0 · · · 0
Therefore one can write
for some matrices A ∈ GL(n, Z) and B ∈ GL(r, Z).
L = B∆A (51)
The numbers di are called the invariant factors of the module, and for a maximal
module one can show that these numbers are all equal to one. The above normal form
result can be proved equivalently by elementary operations on rows and columns or
using the structure of finitely generated modules over a principal domain.
One can easily check that t A sends the standard submodule (which is the one generated
by the first r vectors of the canonical basis of Z n ) to the submodule Λ. So quantitative
information about the submodule is encoded in those matrices A ∈ GL(n, Z).
Following Lochak, we define cΛ (resp. c ′ Λ ) as the minimal value of the norm |A−1 |
(resp. of |A|) among all matrices A ∈ GL(n, Z) satisfying the relation (51) (it is easy
to see that those constants depend only on Λ and not on the choice of such a matrix).
In the space Mn(Z) we may choose the norm | . | induced by the usual supremum norm
for vectors, which is nothing but the maximum of the sums of the absolute values of the
elements in each row.
With these definitions, one can state the following stability result in the Gevrey
class.
Theorem 6.8 (Marco-Sauzin). Let Λ be a K-submodule of Z n of rank r, with r ∈
{0, . . ., n − 1}. Assume that ε ≥ 0 satisfies
εc 5(n−r)
Λ

144 Improved exponential stability for quasi-convex Hamiltonian systems
and H satisfies (C(α, L, M, m, ε)). Then for any solution (θ(t), I(t)) with I0 ∈ B(0, R/2)
and d(ω(0), RΛ)

6.4 - The Gevrey case 145
with the estimates
|u| ≤ |y| |x|
, |v| ≤
d d .
To see this, note that the existence of at least one solution u0, v0 for the above
equation follows easily from the Euclidean division algorithm. Then obviously
u = u0 − k y
d , v = v0 − k x
d ,
is also a solution for any k ∈ Z. Therefore choosing k properly we can find at least one
solution u, v with
|u| ≤ |y|
− 1.
d
For this solution one has
since d − |x| ≤ 0, so
|v||y| = |d − ux|
≤ d + |ux|
≤ d + ( |y|
− 1)|x|
d
≤ |y|
|x| + d − |x|
d
≤ |y|
d |x|
|v| ≤ |x|
d
which proves our claim. Finally, if for instance y = 0, then d = x and one can obviously
choose u = 1 and v = 0.
6.4.0.3. Let K ≥ 1 be given. We can now state our induction hypothesis H(n) for
n ≥ 1.
H(n). Let k = (k1, . . ., kn) be a vector in Z n \ {0} with co-prime components such
that |k|1 ≤ K. Then there exists a matrix A ∈ GL(n, Z) with first row equal to k, which
satisfies |A| ≤ K.
The assertion H(1) is immediate since in this case k = (±1). Now for n ≥ 2,
assume that H(n-1) holds true and consider k = (k1, . . .,kn) in Z n \ {0} with co-prime
components and |k|1 ≤ K.
We may suppose that k∗ = (k1, . . .,kn−1) is non zero (otherwise we consider k∗ =
(k2, . . .,kn)) and we set d = gcd(k1, . . .,kn−1). So d ≥ 1, the integers d −1 k1, . . .,d −1 kn−1
are co-prime, and
|d −1 k∗|1 ≤ K
d .
By H(n-1) we can find a matrix
⎛
⎜
⎝
d −1 k1 · · · d −1 kn−1
l2,1 · · · l2,n−1
.
ln−1,1 · · · ln−1,n−1
.
⎞
⎟ ∈ GL(n − 1, Z),
⎠

146 Improved exponential stability for quasi-convex Hamiltonian systems
such that
for each i ∈ {2, . . ., n − 1}.
n−1
| li,j| ≤ K,
j=1
Now since d and kn are co-prime, one can find integers u, v ∈ Z such that
ud + vkn = 1
and therefore define a matrix
⎛
⎜
A(u, v) = ⎜
⎝
k1
l2,1
.
ln−1,1
· · ·
· · ·
· · ·
kn−1
l2,n−1
.
ln−1,n−1
kn
0
.
0
(−1) n−1 vd −1 k1 · · · (−1) n−1 vd −1 kn−1 (−1) n−1 u
Expanding the determinant along the last column easily proves that A(u, v) ∈ GL(n, Z).
As for the estimates, first assume that kn = 0. Then d = 1 and we may choose u = 1
and v = 0, so obviously
|A(1, 0)| ≤ K.
If now kn is non zero, then by our previous remark we can choose (u∗, v∗) so that
|u∗| ≤ |kn|, |v∗| ≤ d,
which proves that the ℓ 1 –norm of the last row is bounded by K, and therefore
|A(u∗, v∗)| ≤ K. This ends the proof.
With these estimates, one can deduce from Theorem 6.8 the following lemma.
Lemma 6.9. Let Λ be a K-submodule of Zn of rank 1. Assume that ε ≥ 0 and K ≥ 1
satisfy
εK 5(n−1)2

6.4 - The Gevrey case 147
with suitable stable constants K0 and ε0 so that conditions (47) and (52) are satisfied if
0 ≤ ε ≤ ε0, 0 < γ ≤ 5 −1 (n − 1) −2 .
Then we can apply both Lemma 6.4 and Lemma 6.7 in the same way as in the proof
of Theorem 6.2, and setting
aγ =
1 − 5γ(n − 1)2
, bγ =
2α(n − 1)
1 − γ(n − 1)(3n − 1)
,
2(n − 1)
we find that all solutions (θ(t), I(t)) starting at (θ0, I0), with I0 ∈ B(0, R/2), satisfy
Next, using our condition
we have the bounds
hence γ ≤ bγ so that
To conclude, just set
so that
hence
provided that
from which we will only retain
|I(t) − I0|

148 Improved exponential stability for quasi-convex Hamiltonian systems

Part IV
Results of instability
Summary
7 Optimal time of instability for a priori unstable Hamiltonian systems
151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3 Construction of the perturbation . . . . . . . . . . . . . . . . . . . . 155
7.3.1 Integrable system . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.2 Perturbed system . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.4 Construction of a symbolic dynamic . . . . . . . . . . . . . . . . . . 159
7.4.1 Symbolic dynamic . . . . . . . . . . . . . . . . . . . . . . . . 161
7.4.2 Proof of Proposition 7.2 . . . . . . . . . . . . . . . . . . . . . 166
7.5 Construction of a pseudo-orbit . . . . . . . . . . . . . . . . . . . . . 176
7.6 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.A Time-energy coordinates for the pendulum . . . . . . . . . . . . . . . 191
8 Time of instability for high-dimensional Hamiltonian systems 193
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.3 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.3.1 The mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.3.2 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . 204
8.A Gevrey functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Références 209
149

150

7.1 - Introduction 151
7 Optimal time of instability for a priori unstable
Hamiltonian systems
In this chapter, we consider an a priori unstable Hamiltonian system with three degrees
of freedom, for which we construct a drifting solution with an optimal time of diffusion.
Such a result has been already proved by Berti, Bolle and Biasco using variational
arguments, and by Treschev with his separatrix map theory. Our approach is different,
it is based on polysystems, which are a special type of symbolic dynamics corresponding
to the random iteration of a family of maps.
7.1 Introduction
The theory of perturbations of Hamiltonian systems is essentially the study of nearintegrable
Hamiltonian systems, generated by functions of the form
H(θ, I) = h(I) + f(θ, I), (θ, I) ∈ A n = T n × R n ,
where H is sufficiently smooth and f sufficiently small.
7.1.0.1. For f = 0, H = h is an integrable system and the situation is well-understood.
In this case, the variables (θ, I) are called angle-action coordinates for h and the Hamiltonian
depends only on the action variables. The phase space A n is trivially foliated by
invariant Lagrangian tori parametrized by the action variables: indeed the equations of
motion of H read ˙θ = ∇h(I)
˙
I = 0,
so for each I0 ∈ R n , the torus T0 = T n × {I0} is Lagrangian and invariant under the
Hamiltonian flow. The latter is complete and restricts to a linear flow on T0 with
frequency ω0 = ∇h(I0), that is
Φ H t (θ0, I0) = (θ0 + tω0 [Z n ], I0) ∈ T0, t ∈ R.
The action variables remain fixed for all times, and all solutions are quasi-periodic.
7.1.0.2. Now if we let f to be non zero but small, that is
|f| = ε 0, which is called the diffusion time.

152 Optimal time of instability for a priori unstable Hamiltonian systems
7.1.0.3. However, showing that such drifting solutions exist in some large class of nearintegrable
Hamiltonian systems is a very difficult task. Indeed, on the one hand KAM
theory ([Kol54], [Mos62] and [Arn63a]) gives, under some non-degeneracy condition on
h and provided ε is small enough, the existence of a set of positive measure of quasiperiodic
solutions. These solutions are perpetually stable, in the sense that the variation
of their action components is of order at most √ ε for all time. Moreover, for n = 2
and if the integrable Hamiltonian is isoenergetically non degenerate ([Arn63a]), one can
even show that all solutions are stable for all time. On the other hand, for n ≥ 3 we
know from Nekhoroshev theory ([Nek77], [Nek79]) that all solutions are stable, not for
all time, but for an interval of time which is exponentially long with respect to ε −1 ,
provided that h meets some quantitative transversality condition and the perturbation
is sufficiently small.
But even though the existence of drifting orbits for near-integrable Hamiltonian systems
seems to be quite exceptional, Arnold conjectured that this topological instability
is in fact a "typical" phenomenon (see [Arn63a] or [AKN06]). We refer to [Loc99] for a
very lucid and enlightening discussion on Arnold mechanism (and on Arnold diffusion in
general). Under some "generic" conditions and for n = 3, Mather ([Mat04]) announced
that this conjecture holds if the unperturbed Hamiltonian is convex (the convexity is
required by the use of his variational methods). This is so far the best result in this
direction, however, his proof is highly technical and still incomplete. Another connected
question, which is even harder, is to find orbits that densely fill the energy level (this
is the so-called quasi-ergodic hypothesis) : for instance, in [Her98] it is asked whether
there exists a Hamiltonian system, C ∞ -smooth and for r ≥ 2, C r -close to the integrable
Hamiltonian h(I) = 1
2 |I|2 , which has a dense orbit on an energy level (progress towards
this question have been made recently in [KLS10] and [KZZ09]).
7.1.0.4. Yet there is another case, which is much simpler and hence more studied
and understood, in which the unperturbed Hamiltonian is completely integrable (in
the sense of symplectic geometry) but possesses some "hyperbolicity". The prototype
for such a system is given by an uncoupled product of rotators with a pendulum, as
in Arnold’s example. This case is usually referred to as a priori unstable (following
the terminology introduced in [CG94]), in contrast with the a priori stable case where
the unperturbed system is integrable in angle-action coordinates and therefore has no
hyperbolic feature (one can also say that a priori stable systems are "fully elliptic").
The interest in studying such a priori unstable systems is in fact double. First, one can
use the original strategy of Arnold to find unstable orbits, by constructing and following
a "transition chain" made of sets with suitable hyperbolic properties (or minimizing
properties when using variational arguments). The second interest lies in the fact that,
due to normal form theory, the study of a priori stable systems in a neighbourhood
of a simple resonance reduces to the a priori unstable case (see [LMS03] for example).
Results on the topological instability of a priori unstable systems under fairly general
conditions can be found in [DdlLS06] and [Tre04] by means of geometrical methods
and in [CY04], [Ber08] and [CY09] by variational methods. However there are many
difficulties in using these results to tackle Arnold’s conjecture which concerns a priori
stable systems.
7.1.0.5. Another feature of a priori unstable systems is that one avoids all exponentially
small phenomena which are typical for analytic (or Gevrey) a priori stable

7.2 - Main results 153
systems, as the lower bound on the diffusion time imposed by Nekhoroshev’s theorem.
Indeed, if µ denotes the small parameter in the a priori unstable case, then it was
realized by Lochak that the time of diffusion should be polynomial, and then Bernard
([Ber96]), adapting Bessi’s work ([Bes96]), showed that one can obtain an upper bound
on the time of diffusion which is of order µ −2 . In [Loc99], it was conjectured that the
optimal time of diffusion should be µ −1 ln µ −1 , and this was proved by Berti, Bolle and
Biasco [BBB03], still using Bessi’s ideas, and also by Treschev ([Tre04]) and Cresson
and Guillet ([CG03]), using different geometric methods.
The goal of this chapter is to give yet another method to construct an example
with this optimal time of diffusion. Our approach is dynamical: it uses the notion of
polysystem introduced by Moeckel ([Moe02]) in the context of Arnold diffusion, and
which corresponds to the random iteration of a family of maps. More precisely, our
example uses an explicit construction of a polysystem, which is similar to the abstract
mechanism introduced by Marco in [Mar08].
7.2 Main results
7.2.0.1. For n ≥ 1, let us recall that a function f ∈ C ∞ (A n ) is α-Gevrey, for α ≥ 1, if
for any compact subset K ⊆ A n there exist two positive constants AK, BK such that
|∂ k f| C 0 (K) ≤ AKB |k|
K (k!)α , k ∈ N 2n ,
with the standard multi-index notation. We shall denote by G α (A n ) the space of such
functions. For α = 1, these are exactly the analytic functions, but for α > 1, the space
G α (A n ) contains non zero compactly-supported functions (we refer to [MS02], Appendix
A, for more details). Our perturbation will be α-Gevrey for α > 1.
In all this paper, we will consider a Hamiltonian system h with two degrees of freedom
defined by
h(θ1, θ2, I1, I2) = 1
2 (I2 1 + I 2 2) + cos 2πθ1, (θ1, θ2, I1, I2) ∈ A 2 ,
which is the direct product of a pendulum P(θ1, I1) = 1
2I2 1 + cos 2πθ1 on the first factor
and a standard rotator S(θ2, I2) = 1
2I2 2 on the second factor. It is the simplest example
of an a priori unstable integrable Hamiltonian system.
Below we shall state two versions of our theorem, one for Hamiltonian diffeomorphisms
(Theorem 7.2) and one for Hamiltonian flows (Theorem 7.1). For the discrete
case, our unperturbed diffeomorphism is the time-one map of the Hamiltonian flow generated
by h, Φ h : A 2 → A 2 . In the continuous case, our unperturbed system is the a
priori unstable Hamiltonian with three degrees of freedom defined by
ˆh(θ, I) = h(θ1, θ2, I1, I2) + I3 = 1
2 (I2 1 + I2 2 ) + I3 + cos 2πθ1.
7.2.0.2. Let us state our main result.

154 Optimal time of instability for a priori unstable Hamiltonian systems
Theorem 7.1. For α > 1, there exist positive constants C, µ0 and a function f ∈
G α (A 3 ) such that if 0 < µ ≤ µ0, the Hamiltonian system
has an orbit (θ(t), I(t))t∈R such that
with the estimates
H(θ, I) = ˆ h(θ, I) + µf(θ, I), (θ, I) ∈ A 3 ,
lim
t→±∞ I2(t) = ±∞,
|I2(τ) − I2(0)| ≥ 1, τ ≤ Cµ −1 lnµ −1 .
Our unstable orbit is bi-asymptotic to infinity, and the estimates show that the
time of diffusion is of order µ −1 ln µ −1 . As we have already said, similar examples have
been constructed by means of very different methods, in [BBB03], [Tre04] and [CG03].
Moreover, in [BBB03] the authors proved that for such a system the following stability
estimates hold true in the analytic case: given any ρ > 0, there exist positive constants
µ ′ 0 and c such that for µ ≤ µ ′ 0,
|I(t) − I(0)| ≤ ρ, |t| ≤ cµ −1 ln µ −1 .
Therefore this time of diffusion is optimal within the analytic category, but it should
also holds with less regularity, for instance in the Gevrey case ([MS02]) but also in
the C k case for k large enough (see the fifth chapter), as the time of stability is only
polynomial, and in fact almost linear. However, in [MS02] and in the fifth chapter,
only the quasi-convex case has been studied, and to obtain a stability result similar to
[BBB03] one also needs the more general steep case.
Let us also add that even though our perturbation is only Gevrey regular, using the
techniques developed in [LM05] one can obtain Theorem 7.1 in the analytic case, but
with considerably more work.
7.2.0.3. In fact we shall not prove Theorem 7.1 directly, but the following equivalent
equivalent version in terms of diffeomorphisms. Given a function H on A n , we shall
denote by Φ H t : A n → A n the time-t map of its Hamiltonian flow and by Φ H = Φ H 1 the
time-one map.
Theorem 7.2. For α > 1, there exist positive constants C, µ0 and a function f ∈
G α (A 2 ) such that if 0 < µ ≤ µ0, the diffeomorphism
has an orbit (θ k , I k )k∈Z such that
with the estimates
Φ h ◦ Φ µf : A 2 −→ A 2
lim
k→±∞ Ik 2
= ±∞,
|I N 2 − I0 2 | ≥ 1, N ≤ Cµ−1 ln µ −1 .

7.3 - Construction of the perturbation 155
Theorem 7.1 follows easily from Theorem 7.2 by a classical suspension argument (see
[MS02] for a simple method in the Gevrey case using generating functions), so we shall
not repeat the details. Of course, the constants µ0, C and the function f are not the
same in both theorems, but we have kept the same notation for simplicity. Moreover, in
the sequel we will not give explicit values for these constants, in fact sometimes it will
be more convenient to use asymptotic notations: given u(µ) and v(µ) defined for µ ≥ 0,
we shall write u(µ) = O(v(µ)) if there exist positive constants µ ′ and c ′ independent of
µ such that the inequality u(µ) ≤ c ′ v(µ) holds true for 0 ≤ µ ≤ µ ′ .
7.2.0.4. The plan of this chapter is the following. In section 7.3, we will describe
the perturbation. We will show that the perturbed system has an invariant normally
hyperbolic manifold, the stable and unstable manifolds of which intersect transversely
along a homoclinic annulus.
Then, this will be used in section 7.4 to show the following generalisation of the
Birkhoff-Smale theorem to the normally hyperbolic case: near the homoclinic manifold,
there exist an invariant set on which a suitable iterate of the system is conjugated to a
symbolic dynamic, more precisely to a skew-product on the annulus A over a Bernoulli
shift. Let us give a precise definition.
Given an alphabet A ⊆ N, we let ΣA = A Z be the Cantor set of bi-infinite sequence
of elements in A. We will denote by σ = σA the left Bernoulli shift on ΣA, that is if
¯n = (nk)k∈Z belongs to ΣA, then σ(¯n) = (n ′ k )k∈Z is defined by
n ′ k = nk+1, k ∈ Z.
Definition 7.3. Let M be a manifold. A skew-product on M over σ is a map G :
ΣA × M −→ ΣA × M of the form
G(¯n, x) = (σ(¯n), F¯n(x)), x ∈ M,
where F¯n : M → M is an arbitrary map, for ¯n ∈ ΣA.
However we are not able to prove the existence of our orbit using directly this symbolic
dynamic. Hence in section 7.5, we will first construct this orbit for a simplified
model called polysystem, which appears as a random iteration of "standard" maps of
the annulus, and which is close to our skew-product. Here is the definition.
Definition 7.4. A polysystem on M is a skew-product over a Bernoulli shift of the form
with fn : M → M for n ∈ A.
F¯n(x) = fn0(x), ¯n ∈ ΣA, x ∈ M,
Then in section 7.6, the orbit for the polysystem will be considered as a pseudo-orbit
for the skew-product and we will conclude using shadowing arguments. Finally we have
gathered in an appendix some estimates on the so-called time-energy coordinates for
the simple pendulum that are used in section 7.4.
7.3 Construction of the perturbation
This section is devoted to a description of our system, which is similar to the one
introduced in [Mar05]. We will first consider the integrable case, and then explains the
construction of the perturbation.

156 Optimal time of instability for a priori unstable Hamiltonian systems
I1
O •
7.3.1 Integrable system
θ1
S
×
I2
Φ P Φ S
Figure 1: Integrable diffeomorphism F0
Our integrable diffeomorphism F0 : A 2 → A 2 is the time-one map of the Hamiltonian
flow generated by h, that is
F0 = Φ h = Φ 1
2 (I2 1 +I2 2 )+cos 2πθ1 ,
which is the product of the pendulum map Φ P = Φ 1
2 I2 1 +cos 2πθ1 and the integrable twist
map Φ S = Φ 1
2 I2 2 (see figure 1).
It is an "a priori unstable" map in the sense that it possesses an invariant normally
hyperbolic annulus. Indeed, let O = (0, 0) be the hyperbolic fixed point of the pendulum
map Φ P , its stable and unstable manifolds obviously coincide, and if S is the upper part
of the separatrix, then
W ± (O, Φ P ) = S = {(θ1, I1) ∈ A | I1 = 2 sin πθ1}.
Due to the product structure of the map F0, it is easy to see that the annulus A = O×A
is invariant by F0, and one can check that it is symplectic for the canonical structure of
A 2 . But the most important feature is that this annulus is normally hyperbolic, more
precisely it is r-normally hyperbolic for any r ∈ N, in the sense of [HPS77]. To see this,
just decompose the tangent bundle of A 2 along A as
TAA 2 = T A ⊕ E s ⊕ E u ,
where E s (resp. E u ) is the one-dimensional contracting (reps. expanding) direction
associated to the hyperbolic fixed point. Then note that the restriction of F0 to A
coincides with the integrable twist map Φ S , hence there is zero contraction and expansion
in the tangent direction T A.
The 3-dimensional stable and unstable manifolds of A also coincide, and are given
by the product
W ± (A, F0) = S × A.
θ2

7.3 - Construction of the perturbation 157
7.3.2 Perturbed system
Now we will describe the geometric properties of our perturbed system. It will be of the
form
Fµ = Φ µf ◦ F0,
where µ > 0 is the small parameter, and f : A 2 → R a function of the form
to be defined precisely below.
f(θ1, θ2, I1, I2) = χ(θ1)f1(θ1, I1)f2(θ2)
7.3.2.1. First χ : T → R will be a bump function. Of course such a function can be
chosen to be α-Gevrey for α > 1 but not analytic. More precisely, let us choose ¯ θ ∈ T
such that
( ¯ θ, I) ∈ S =⇒ Φ P ( ¯ θ, I) = (1 − ¯ θ, I).
Since the separatrix S is symmetric about the section {θ1 = 1/2}, ¯ θ is well-defined.
Then choose any ˜ θ ∈ ] ¯ θ, 1/2[, and a function χ ∈ G α (T), α > 1, such that
χ(θ) =
1 if θ ∈ [ ˜ θ, 1 − ˜ θ],
0 if θ /∈ [ ¯ θ, 1 − ¯ θ].
Since χ is identically zero at 0, the perturbation Φ µf is the identity on the normally
hyperbolic annulus A, and therefore the latter remains invariant and normally hyperbolic
for the map Fµ. Moreover, as χ vanishes in a neighbourhood of 0, some pieces of the
stable and unstable manifolds for Fµ will coincide with the ones of F0.
7.3.2.2. Now to define the function f1 : A → R, we will use time-energy coordinates
on the first factor, so we introduce the symplectic diffeomorphism
where
and
Ψ : E × A −→ E∗ × A
(θ1, I1, θ2, I2) ↦−→ (τ, e, θ2, I2)
E = {(θ1, I1) ∈ A | 0 < θ1 < 1, I1 > 0}
E∗ = {(τ, e) ∈ R 2 | e > −2, |τ| < 1
T(e)}. (53)
2
We refer to Appendix 7.A for more details on those coordinates. The function f1 is
simply defined by
f1(τ, e) = 1 − 1
2 τ2 , (τ, e) ∈ E∗,
and it will create a transverse intersection between the stable and unstable manifolds
W + (A, Fµ) and W − (A, Fµ).
7.3.2.3. Finally the function f2 will be defined by
f2(θ2) = −π −1 (2 + sin 2πθ2), θ2 ∈ T.

158 Optimal time of instability for a priori unstable Hamiltonian systems
The fact the f2 is nowhere zero on T (for any θ2 ∈ T, |f2(θ2)| ≥ π −1 ) will imply that
W + (A, Fµ) and W − (A, Fµ) intersect transversely along an annulus, and the explicit
form of f2 will be responsible for the non trivial dynamics along this homoclinic annulus.
7.3.2.4. Let us define the domains
D = ([ ¯ θ, 1 − ¯ θ] × R + ∗ ) × A ⊆ A 2 , D∗ = Ψ(D) ⊆ E∗ × A,
on which the perturbation (either in the original coordinates or in time-energy coordinates)
is non zero, and
D = ([ ˜ θ, 1 − ˜ θ] × R + ∗ ) × A ⊆ A 2 , D∗ = Ψ( D) ⊆ E∗ × A.
For (τ, e, θ2, I2) ∈ D∗, our perturbation can be written explicitly as
Φ µf (τ, e, θ2, I2) = (τ, e − µf ′ 1 (τ)f2(θ2), θ2, I2 − µf1(τ)f ′ 2 (θ2))
and otherwise the diffeomorphism Φ µf is the identity.
= (τ, e + µτf2(θ2), θ2, I2 + 2µf1(τ) cos(2πθ2)), (54)
7.3.2.5. In the sequel we shall need the following property.
Proposition 7.1. The immersed manifolds W + (A, Fµ) and W − (A, Fµ) intersect transversely
along the annulus
Iµ = {(τ, e, θ2, I2) ∈ E∗ × A | τ = e = 0}.
Let us remark that this annulus is also given by
in the original coordinates.
Iµ = {(θ1, I1, θ2, I2) ∈ A 2 | θ1 = 1/2, I1 = 2}
Proof. By definition of D, one easily sees that the sets
W + (A, F0) ∩ F0( D), W − (A, F0) ∩ F −1
0 ( D),
are disjoint from D. Since Fµ and F0 coincide outside D, we can define pieces of stable
and unstable manifolds by
and
Therefore
W + ¯ θ (A, Fµ) = W + (A, F0) ∩ F0( D) ⊆ W + (A, Fµ)
W − ¯ θ (A, Fµ) = W − (A, F0) ∩ F −1
0 ( D) ⊆ W − (A, Fµ).
F −1
µ (W + ¯ θ (A, Fµ)) ⊆ W + (A, Fµ), Fµ(W − ¯ θ (A, Fµ)) ⊆ W − (A, Fµ). (55)
Then on the one hand,
F −1
µ (W + ¯ θ (A, Fµ)) = F −1
0 ◦ Φ−µf (W + ¯ θ (A, Fµ))
= F −1
0 (W + ¯ θ (A, Fµ))
= W + (A, F0) ∩ D,

7.4 - Construction of a symbolic dynamic 159
and on the other hand,
Fµ(W − ¯ θ (A, Fµ)) = Φ µf ◦ F −1
0 (W − ¯ θ (A, Fµ))
Now in time-energy coordinates, one simply has
and therefore
= Φ µf (W − (A, F0) ∩ D).
W ± (A, F0) = {(τ, e, θ2, I2) ∈ E∗ × A | e = 0},
W + (A, F0) ∩ D = {(τ, e, θ2, I2) ∈ D∗ | e = 0},
while, using the expression of the perturbation (54),
Φ µf (W − (A, F0) ∩ D) = {(τ, e, θ2, I2) ∈ D∗ | e = µτf2(θ2)}.
Since f2 is nowhere zero, one easily sees that the manifolds F −1
µ (W + ¯ θ (A, Fµ)) and
Fµ(W − ¯ θ (A, Fµ)) intersect transversely along the annulus
and the conclusion follows by (55).
Iµ = {(τ, e, θ2, I2) ∈ D∗ | τ = e = 0},
7.4 Construction of a symbolic dynamic
7.4.0.1. In this section, we will take advantage of the fact that our diffeomorphism
Fµ : A 2 → A 2
possesses an invariant normally hyperbolic annulus A, whose stable and unstable manifolds
intersect transversely along a homoclinic annulus Iµ.
In the case where the normally hyperbolic manifold is a point, under such a transverse
homoclinic intersection it is well-known that chaotic dynamics arise: the system has an
invariant Cantor set on which a suitable iterate is conjugated to a shift map (this is the
Horseshoe theorem, due to Birkhoff, Smale and Alexeiev). In our more general situation,
the symbolic dynamic is more complicated.
7.4.0.2. Recall from definition 7.3 that a skew-product (over σ) is completely defined
by the family of maps (F¯n)¯n∈ΣA : M → M, and the skew-product will be denoted by
[[F¯n]]n∈ΣA . Then one can easily see that a sequence (nk, xk)k∈Z ∈ A × M gives rise to
an orbit (σk (¯n), xk)k∈Z ∈ ΣA × M, where ¯n = (nk)k∈Z, for the skew-product [[F¯n]]n∈ΣA
if and only if
Fσk (¯n)(xk) = xk+1, k ∈ Z.
In this section, we will need our alphabet A to contain integers of order ln µ −1 , but
for subsequent arguments, we will also need it to contain integers n ∈ N as large as
µ −1
2 ln µ −1 , so we may already fix
A = Aµ = {[ln µ −1 ], . . .,2[µ −1
2 ln µ −1 ]}.

160 Optimal time of instability for a priori unstable Hamiltonian systems
For simplicity, we shall get rid of the subscript µ.
7.4.0.3. Let Oµ ⊆ D∗ be a neighbourhood (in time-energy coordinates) of the homoclinic
annulus Iµ. For x ∈ Oµ, let us set
and assuming n x 0
n x k
n x 0 = inf{n ∈ N∗ | F n µ (x) ∈ Oµ} ∈ N ∗ ∪ {+∞}
< +∞, we can define inductively
= inf n ∈ N ∗ | F n
µ F nx
k−1
µ (x)
∈ Oµ
for k ≥ 1 provided nx k−1 < +∞. Similarly we can define
and by induction
if n x −k
n x −k
∈ N ∗ ∪ {+∞},
n x −1 = inf{n ∈ N∗ | F −n
µ (x) ∈ Oµ} ∈ N ∗ ∪ {+∞},
= inf
< +∞ for k ≥ 1.
n ∈ N ∗ | F n
µ
F nx −k+1
µ
(x) ∈ Oµ ∈ N ∗ ∪ {+∞}
We will show below that there exists Λµ ⊆ Oµ such that for any x ∈ Λµ, the
doubly infinite sequence ¯n x = (n x k )k∈Z is a well-defined element of ΣA. In fact, Λµ is
homeomorphic to a Cantor set of annuli, and we will be interested in the dynamics
restricted to this set. For that, following Moser ([Mos73]) we define the transversal map
˜Fµ(x) = F nx 0
µ (x), x ∈ Oµ, n x 0 < +∞,
which is the first return map to the neighbourhood Oµ, and if the sequence ¯n x is welldefined,
then so is ˜ F n µ(x) for n ∈ Z.
7.4.0.4. In the sequel, we will consider the discrete topology on the alphabet A, and
the sets A N and ΣA = A Z will be endowed with the product topology, for which they are
compact and metrizable. The goal of this section is to prove the following proposition.
Proposition 7.2. For µ small enough, there exist a neighbourhood Oµ, a set Λµ ⊆ Oµ
invariant by the transversal map ˜ Fµ, a homeomorphism
Υµ : ΣA × A −→ Λµ
(¯n, (θ, I)) ↦−→ (ψ¯n(θ, I), (θ, I))
with ψ¯n : A → E∗, where E∗ is defined in (53), and a family of maps
F¯n(θ, I) = (θ + n0I, I + 2µf1(φ¯n(θ, I)) cos2π(θ + n0I)), ¯n ∈ ΣA, (θ, I) ∈ A,
such that for x = (τx, ex, θx, Ix) ∈ Λµ, then ¯n x ∈ ΣA and
˜Fµ ◦ Υµ(¯n x , θx, Ix) = Υµ(σ(¯n x ), F¯n x(θx, Ix)).
Moreover the map φ¯n : A → R is Lipschitz and satisfies
|φ¯n|C 0 (A) = O µ 2π−1 ,
Lip(φ¯n) = O µ 2π−1 2 ln µ −1
. (56)

7.4 - Construction of a symbolic dynamic 161
The statement of the above proposition seems complicated, but in fact it simply
means that the restriction of ˜ Fµ to the invariant set Λµ is conjugated to the skewproduct
on A given by the maps
F¯n(θ, I) = (θ + n0I, I + 2µf1(φ¯n(θ, I)) cos 2π(θ + n0I)), ¯n ∈ ΣA, (θ, I) ∈ A.
The proof of this result is long and technical. We will first prove an abstract result
in section 7.4.1, which is contained in Proposition 7.4, and then we will apply this
proposition to our example in section 7.4.2.
7.4.1 Symbolic dynamic
7.4.1.1. Here we will use the framework developed by Chaperon ([Cha04], [Cha08]),
which was designed to obtain rather general invariant manifold theorems, in particular
in the normally hyperbolic case.
Consider a complete metric space X and a complete subspace Y of a metric space F.
We endow the product space X ×F with the product metric, that is for (x, y), (x ′ , y ′ ) ∈
X × F we define
d((x, y), (x ′ , y ′ )) = sup{d(x, x ′ ), d(y, y ′ )}.
Let us set Z = X × Y . Then we consider a Lipschitz map
h = (f, g) : Z −→ X × F,
and we make the following two assumptions.
(H1) There exists a constant ρ −1
0 > 0 such that for all x ∈ X, y, y′ ∈ Y
Hence, for all x ∈ X, the map
is a bijection.
d(g(x, y), g(x, y ′ )) ≥ ρ −1
0 d(y, y′ ).
gx : Y −→ gx(Y )
(H2) For all x ∈ X, we have Y ⊆ gx(Y ). Hence the map
is well-defined, and if
g −1
x : Y −→ Y
G : Z −→ Y
is defined by G(x, y) = g−1 x (y), then G is Lipschitz.
We will say that the map h satisfies hypothesis (H) if it satisfies both hypotheses
(H1) and (H2). Under these assumptions, we will consider three positive constants ν0, σ0
and κ0 where ν0 is the Lipschitz constant of G with respect to Y , that is
d(G(x, y), G(x, y ′ )) ≤ ν0d(y, y ′ ), ∀x ∈ X, ∀y, y ′ ∈ Y,

162 Optimal time of instability for a priori unstable Hamiltonian systems
which from (H1) is smaller than ρ0, σ0 is the Lipschitz constant of G with respect to X,
that is
d(G(x, y), G(x ′ , y)) ≤ σ0d(x, x ′ ), ∀x ∈ X, ∀y, y ′ ∈ Y,
and κ0 the Lipschitz constant of f. Let us now formulate another hypothesis.
(L) Assume that σ0 + ν0 max{κ0, 1} < 1. Hence for any x ∈ X, y ∈ Y the maps
are contractions.
G(., y) : X −→ Y, G(x, .) : Y −→ Y,
The following result is due to Chaperon ([Cha04]).
Theorem 7.5 (Chaperon). With the previous notations, assume that Y is bounded and
the map
h : Z −→ X × F
satisfies hypotheses (H) and (L). Then the set
Λ =
h −n (Z)
n∈N
is the graph of a contracting map Φ : X → Y , with
Lip(Φ) ≤ σ0(1 − ν0κ0) −1 < 1.
7.4.1.2. The above theorem is concerned with the iteration of a single map h, but
with this formalism we obviously have an analogous result for the iteration of a family
of maps. More precisely, consider a family of maps
hn = (fn, gn) : Z −→ X × F, n ∈ A,
where A is the given alphabet. We will assume that each map satisfies hypothesis (H)
and a uniform version of hypothesis (L), that is:
(L’) sup n∈A {σn + νn max{κn, 1}} < 1.
Then one can state the following proposition.
Proposition 7.3. With the previous notations, assume that Y is bounded and that for
any n ∈ A the maps
hn : Z −→ X × F
satisfy hypotheses (H) and (L’). Then for any sequence ¯n+ = (nk)k∈N ∈ AN , the set
Λ + (¯n+) =
(hnk ◦ · · · ◦ hn0) −1 (Z)
k∈N
is the graph of a contracting map Φ¯n+ : X → Y , with
Lip(Φ¯n+) ≤ sup{σn(1
− νnκn)
n∈A
−1 } < 1.
Moreover, if we endow the space of continuous function C(X, Y ) with the topology of
pointwise convergence, then the map
is continuous.
¯n+ ∈ A N ↦−→ Φ¯n+ ∈ C(X, Y )

7.4 - Construction of a symbolic dynamic 163
The proof is the same as in Theorem 7.5, with some obvious modifications.
7.4.1.3. Now we consider two metric spaces F + and F − , and two complete subspaces
X ⊆ F − and Y ⊆ F + . Let V another complete metric space, and
Z = X × Y × V.
One has to think of V as a central direction and F − (resp. F + ) as a contracting (resp.
expanding) direction. We consider two families of maps (h + n)n∈A and (h − n)n∈A of the
form
h + n : Z −→ X × F + × V, h − n : Z −→ F − × Y × V,
and we decompose them as h ± n = (f ± n , g± n ), where
and
The hypotheses (H) and (L’) for h ± n
by
f + n : Z −→ X × V, f − n : Z −→ Y × V
g + n : Z −→ F + , g − n : Z −→ F − .
refer to these decompositions. Let us also denote
F + n : Z −→ V
the second component of the map f + n , and consider sets Z + n ⊆ Z and Z− n ⊆ Z such that
for any n ∈ A.
Z ∩ (h + n )−1 (Z) ⊆ Z + n , Z ∩ (h− n )−1 (Z) ⊆ Z − n ,
7.4.1.4. The aim of this section is to prove the following result.
Proposition 7.4. With the previous notations, assume that X, Y are bounded, V is
locally compact and that for any n ∈ A the maps
h + n : Z −→ X × F + × V, h − n : Z −→ F − × Y × V
satisfy hypotheses (H) and (L’). Let us also assume that for any n ∈ A,
(i) the maps (h ± n ) |Z ± n are invertible and
(h + n ) |Z + −1 = (h n
− n ) |Z + n ,
(ii) the sets Z + n (resp. Z− n ) are pairwise disjoint.
(h − n ) |Z − −1 = (h n
+ n ) |Z − n ;
Then for any sequence ¯n ∈ ΣA, the map h + n0 has an invariant set Λ and there exist a
homeomorphism
Υ : ΣA × V −→ Λ
(¯n, v) ↦−→ (Φ¯n(v), v)
which conjugates h +
n0|Λ
to the skew-product on V given by
F¯n(v) = F + n0 (Φ¯n(v), v), ¯n ∈ ΣA, v ∈ V,
where Φ¯n : V → X × Y is a contracting map, with
Lip(Φ¯n) ≤ sup{(σ
n∈A
+ n (1 − ν + n κ + n) −1 ), (σ − n (1 − ν − n κ − n) −1 )} < 1.

164 Optimal time of instability for a priori unstable Hamiltonian systems
As a consequence, the functions F¯n : V → V , for ¯n ∈ ΣA, are also defined by the
following equation
h + n0 (Φ¯n(v), v) = (Φ¯n(F¯n(v)), F¯n(v)), v ∈ V. (57)
This remark will be useful later on to obtain the estimates (56) in Proposition 7.4.
Proof. For ¯n ∈ ΣA, let us define
¯n+ = (nk)k∈N ∈ A N , ¯n− = (n−k)k∈N∗ ∈ AN∗.
Since each family of maps (h + n)n∈A and (h − n)n∈A satisfies hypotheses (H) and (L’), we
can apply Proposition 7.3 so both sets
and
are graphs of contracting maps
that is
and
Moreover, one has
Λ + (¯n+) =
k∈N
Λ − (¯n−) =
k∈N ∗
Φ + ¯n+ : X × V −→ Y, Φ− ¯n−
(h + nk ◦ · · · ◦ h+ n0 )−1 (Z)
(h − n−k ◦ · · · ◦ h− n−1 )−1 (Z)
: Y × V −→ X,
Λ + (¯n+) = {(x, Φ + (x, v), v) | x ∈ X, v ∈ V }
¯n+
Λ − (¯n−) = {(Φ − (y, v), y, v) | y ∈ Y, v ∈ V }.
¯n−
Lip(Φ + ¯n+
Lip(Φ − ¯n−
Therefore for each v ∈ V , the maps
) ≤ sup{σ
n∈A
+ n (1 − ν+ n κ+ n )−1 } < 1, (58)
) ≤ sup{σ
n∈A
− n (1 − ν− n κ−n )−1 } < 1. (59)
Φ + ¯n+,v = Φ + ¯n+ (., v) : X −→ Y, Φ−¯n−,v = Φ − ¯n− (., v) : Y −→ X,
are also contracting, and so are the maps
Φ − ¯n−,v ◦ Φ+ ¯n+,v : X −→ X, Φ+ ¯n+,v ◦ Φ− ¯n−,v
: Y −→ Y.
Since X and Y are complete, these maps have fixed points x(¯n, v) ∈ X and y(¯n, v) ∈ Y
from the contraction principle, and by uniqueness they satisfy
Now let us define
Φ + ¯n+ (x(¯n, v), v) = y(¯n, v), Φ− (y(¯n, v), v) = x(¯n, v).
¯n−
Λ(¯n) = Λ + (¯n+) ∩ Λ − (¯n−).
Then by the previous relation this set is non-empty since it is the graph of the contraction
Φ¯n : V −→ X × Y
v ↦−→ (x(¯n, v), y(¯n, v)),

7.4 - Construction of a symbolic dynamic 165
and, from (58) and (59),
Moreover, as the maps
Lip(Φ¯n) ≤ sup{(σ
n∈A
+ n (1 − ν+ n κ+ n )−1 ), (σ − n (1 − ν− n κ−n )−1 )} < 1.
¯n+ ∈ A N ↦→ Φ + ¯n+ ∈ C(X × V, Y ), ¯n− ∈ A N∗
↦→ Φ − ¯n−
∈ C(Y × V, X),
are continuous, from the contraction principle one also has the continuity of the map
Now set
¯n ∈ ΣA ↦−→ Φ¯n(v) ∈ X × Y.
Λ =
¯n∈ΣA
Λ(¯n).
The fact that this set is invariant under h + will follow from our condition (i). Indeed,
n0
if z ∈ Λ, then z ∈ Λ(¯n) for some ¯n ∈ ΣA and so by definition
z ∈
(h + nk ◦ · · · ◦ h+ n0 )−1 (Z), z ∈
(h − n−k ◦ · · · ◦ h−n−1 )−1 (Z).
k∈N
From the first relation we get
and since by hypothesis
k∈N ∗
h +
n0 (z) ∈
k∈N∗ (h + nk ◦ · · · ◦ h+ n1 )−1 (Z) = Λ + (σ(¯n)+), (60)
(h + n0 ) |Z + −1 = (h n0 − n0 ) |Z + n0 and Λ(¯n) ⊆ Z + n0 , we get from the second relation
h +
(z) ∈ n0
k∈N
(h − n−k ◦ · · · ◦ h− n0 )−1 (Z) = Λ − (σ(¯n)−). (61)
Now (60) and (61) means exactly that h + n0 (z) ∈ Λ(σ(¯n)), so Λ is positively invariant
under h + . In fact, a completely similar argument using condition (i) shows that
n0
h + (Λ(¯n)) = Λ(σ(¯n)),
n0
and hence Λ is totally invariant under h + . More generally, one obtains
n0
h ± nk−1 ◦ · · · ◦ h± n0 (Λ(¯n)) = Λ(σ±k (¯n)), k ∈ Z.
Next let us prove that as a consequence of (ii) the union
Λ =
Λ(¯n)
¯n∈ΣA
is disjoint. Let ¯n = ¯n ′ , so there exists l ∈ Z such that nl = n ′ l . First suppose that l = 0,
then on the one hand
Λ(¯n) ⊆ Λ(¯n+) ⊆ Z + n0

166 Optimal time of instability for a priori unstable Hamiltonian systems
and on the other hand
Λ(¯n ′ ) ⊆ Λ(¯n ′ +
) ⊆ Z+ .
As Z + and Z+ n0 n ′ are disjoint by hypothesis, then so are Λ(¯n) and Λ(¯n
0
′ ). Now if l ≥ 1,
we can assume without loss of generality that nk = n ′ k for 0 ≤ k ≤ l − 1, and as before
and
h + nl−1 ◦ · · · ◦ h+ n0 (Λ(¯n)) = Λ(σl (¯n)) ⊆ Λ(σ l (¯n)+) ⊆ Z + nl
h + nl−1 ◦ · · · ◦ h+ n0 (Λ(¯n′ )) = Λ(σ l (¯n ′ )) ⊆ Λ(σ l (¯n ′ )+) ⊆ Z +
n ′ ,
l
so Λ(¯n) and Λ(¯n ′ ) have to be disjoint. Finally, the case l ≤ −1 is completely similar
using the hypothesis that Z− n are pairwise disjoint for n ∈ A.
To conclude, as Λ is a disjoint union, every point z ∈ Λ can be uniquely written as
z = (Φ¯n(v), v) for v ∈ V and ¯n ∈ ΣA, so the map
n ′ 0
Υ : ΣA × V −→ Λ
(¯n, v) ↦−→ (Φ¯n(v), v)
is a well-defined continuous bijection, and as V is locally compact, one can check that
it is a homeomorphism with respect to the product topology on ΣA × V . Then for
z = (Φ¯n(v), v) ∈ Λ(¯n), as h + n0 (z) ∈ Λ(σ(¯n)) we have from the definition of F + n ,
h + n0 (z) = Φσ(¯n)(F + n0 (z)), F + n0 (z)
The last equality exactly means that the map
is conjugated by Υ to the skew-product
= Φσ(¯n)(F + n0 (Φ¯n(v), v)), F + n0 (Φ¯n(v), v) .
h + : Λ −→ Λ
n0
G : ΣA × V −→ ΣA × V
(¯n, v) ↦−→ (σ(¯n), F(¯n, v)).
where F(¯n, v) = F + n0 (Φ¯n(v), v). This ends the proof.
7.4.2 Proof of Proposition 7.2
This section is entirely devoted to the proof of Proposition 7.2, this will be done in several
steps. We will have to use notations and estimates on time-energy coordinates, contained
in Appendix 7.A, and this will require to choose µ sufficiently small. Moreover, we shall
use various coordinates but we shall keep the same notation for the diffeomorphisms
expressed in different coordinates.
Step 1. Straightening of the invariant manifolds.
Using time-energy coordinates on the first factor, our homoclinic annulus is given by
Iµ = {(τ, e, θ2, I2) ∈ D∗ | τ = e = 0}

7.4 - Construction of a symbolic dynamic 167
and in a neighbourhood of it, the stable manifold of Fµ is given by
while the unstable manifold is
{(τ, e, θ2, I2) ∈ D∗ | e = 0}
{(τ, e, θ2, I2) ∈ D∗ | e = µf2(θ2)τ}.
Now we introduce the change of coordinates
where
Its inverse is given by
Θ : E∗ × A −→ E ′ ∗ × A
(τ, e, θ2, I2) ↦−→ (τ ′ , e ′ , θ2, I2)
τ ′ = τ − (µf2(θ2)) −1 e, e ′ = (µf2(θ2)) −1 e.
Θ −1 : E ′ ∗ × A −→ E∗ × A
(τ ′ , e ′ , θ2, I2) ↦−→ (τ ′ + e ′ , µf2(θ2)e ′ , θ2, I2).
It follows that in these new coordinates, denoting by D ′ ∗ = Θ( D∗), the stable manifold
and the unstable manifold
are straightened out.
Step 2. Choice of the box Z.
{(τ ′ , e ′ , θ2, I2) ∈ D ′ ∗ | e′ = 0}
{(τ ′ , e ′ , θ2, I2) ∈ D ′ ∗ | τ ′ = 0}
Our goal is to use Proposition 7.4, so we will explain how to choose the domain Z.
Our central direction V = A will be the annulus given by the coordinates (θ2, I2), our
contracting and expanding directions will be one-dimensional, so X ⊆ F − = R and
Y ⊆ F + = R. We will choose
X = [−τ ′ 0 , τ ′ 0 ], Y = [0, e′ 0 ],
with τ ′ 0 = c1µ 2π−1 , e ′ 0 = c2µ 2π−1 , and c1 > 2π, c2 > 2π. Eventually
Z = [−τ ′ 0, τ ′ 0] × [0, e ′ 0] × A.
For (θ2, I2) ∈ A, let us also define the section
Z(θ2, I2) = [−τ ′ 0, τ ′ 0] × [0, e ′ 0] × {θ2, I2},
where all the constructions will take place (see figure 2).
This domain Z is located above the homoclinic annulus Iµ, and for µ small enough
it is contained in the domain D ′ ∗ where the manifolds are straightened.

168 Optimal time of instability for a priori unstable Hamiltonian systems
µf2(θ2)e ′ 0
e
−τ ′ 0 τ ′ 0
e = µf2(θ2)τ
Figure 2: Section Z(θ2, I2) of the domain Z
Z(θ2, I2)
For n ∈ A = {[ln µ −1 ], . . .,2[µ −1
2 ln µ −1 ]}, the point an = (0, en) ∈ D∗, or a ′ n =
(−e ′ n, e ′ n) ∈ D ′ ∗, is by definition n-periodic for the pendulum map ΦP . We want the
annulus {a ′ n } × A, which is n-periodic for the unperturbed map F0, to be included in
our domain Z, and this requires that e ′ n < e′ 0 and −τ ′ 0 < −e′ n for any n ∈ A. First note
that
e0 = µf2(θ2)e ′ 0 = c2f2(θ2)µ 2π ,
and as f2(θ2) ≥ π −1 , c2f2(θ2) > 2 so e0 > 2µ 2π . Then using (73) and the fact that
A = {[ln µ −1 ], . . ., 2[µ −1
2 ln µ −1 ]}, for µ is small enough (and hence n is large enough)
one obtains
sup{en}
< 2µ
n∈A
2π ,
and therefore en < e0 is satisfied for any n ∈ A, which gives e ′ n < e′ 0 . Similarly, we
obtain
−τ ′ 0 < −2πµ 2π−1 < −e ′ n.
Step 3. Construction of the domains Z + n .
For n ∈ A, the domains Z + n ∩ Z(θ2, I2) will be rectangles (see figure 3, they are
parallelograms in the coordinates (τ, e)), with vertices
A 1 n =
−τ ′ 0 , en − δe n
µf2(θ2) , θ2,
I2 , A 2 n =
−τ ′ 0 , en + δe n
µf2(θ2) , θ2,
I2 ,
A 3 n =
where the parameter δ e n
τ ′ 0, en + δ e n
µf2(θ2) , θ2, I2
is defined by
δ e n
, A 4
n = τ ′ 0, en − δe n
µf2(θ2) , θ2, I2
2π−1 c3µ
=
|T ′ n |
, n ∈ A
for some constant c3 > max{c1 + c2, c1 + 2π}, and T ′ n = T ′ (en) (see Appendix 7.A
and (74)). Hence the domains Z + n are small neighbourhoods of the annuli {a′ n } × A, for
an = (−e ′ n , e′ n ) ∈ D ′ ∗ .
,
τ

7.4 - Construction of a symbolic dynamic 169
A 2 n
A 1 n
e
A 3 n
A 4 n
Z + n
Figure 3: Section Z + n (θ2, I2) of the domain Z + n
For µ small enough, one can easily check from the definition of δ e n
value theorem that
and
n + c3µ 2π−1 ≤ T(en − δ e 2π−1
n ) ≤ n + 2c3µ
τ
and the mean
(62)
n − 2c3µ 2π−1 ≤ T(en + δ e n ) ≤ n − c3µ 2π−1 . (63)
Let us show that these domains Z + n
to prove that e ′ (A1 n) > e ′ (A2 n+1). Choosing µ small so
from (62) and (63) we get
are pairwise disjoint. By construction it is enough
n + 2c3µ 2π−1 < n + 1 − 2c3µ 2π−1 ,
T(en − δ e n) < T(en+1 + δ e n+1).
But as the period function T is decreasing, this gives
which implies that e ′ (A 1 n) > e ′ (A 2 n+1).
en − δ e n > en+1 + δ e n+1
Let us also remark that by definition of our domain D ′ ∗ , one can ensure that for µ
small enough (and so n − T(e) is small enough)
F k 0 (Z+ n ) ∩ D′ ∗
Step 4. Construction of the domains Z − n .
= ∅, 0 < k < n. (64)
The construction is similar (see figure 4), namely Z− n ∩ Z(θ2, I2) is a rectangle with
vertices
B 1 n =
− en
µf2(θ2) − δτ n, e ′
0, θ2, I2 , B 2
n = − en
µf2(θ2) + δτ n, e ′
0, θ2, I2 ,

170 Optimal time of instability for a priori unstable Hamiltonian systems
B 3 n =
Z − n
where δ τ n is defined by
B 4 n
e
B 3 n
B 1 n B 2 n
Figure 4: Section Z − n (θ2, I2) of the domain Z − n
− en
µf2(θ2) + δτ
n, 0, θ2, I2 , B 2
n = − en
µf2(θ2) − δτ
n, 0, θ2, I2 ,
δ τ n
2π−2 c4µ
=
|T ′ n |
,
for a constant c4 > π max{c1+4π, c1+c2}, and T ′ n = T ′ (en) (see Appendix 7.A and (74)).
As in the previous step, one can check that τ ′ (B 1 n+1 ) > τ ′ (B 2 n ) so the domains Z− n are
pairwise disjoint.
Step 5. Expressions of the maps F ±n
µ restricted to Z ± n .
For any (τ, e, θ2, I2) ∈ Θ −1 (Z + n ), one has the following explicit expression for the
unperturbed map
F n 0 (τ, e, θ2, I2) = (τ + n − T(e), e, θ2 + nI2, I2).
In those coordinates, our perturbation is given by
Φ µf (τ, e, θ2, I2) = (τ, e + µτf2(θ2), θ2, I2 − µf1(τ)f ′ 2(θ2)).
Then, from (64) we know that F n µ = Φ µf ◦ F n 0 when restricted to Θ −1 (Z + n ), so
F n µ(τ, e, θ2, I2) = (τ + n − T(e), e + µ(τ + n − T(e))f2(θ2 + nI2),
θ2 + nI2, I2 − µf1(τ + n − T(e))f ′ 2 (θ2 + nI2)).
Using the expression for Θ −1 , for (τ ′ , e ′ , θ2, I2) ∈ Z + n we compute
F n µ(τ ′ , e ′ , θ2, I2) = (−(f2(θ2 + nI2)) −1 f2(θ2)e ′ ,
(1 + f2(θ2 + nI2)) −1 f2(θ2))e ′ + τ ′ + n − T(µf2(θ2)e ′ ),
θ2 + nI2,
I2 − µf1(τ ′ + e ′ + n − T(µf2(θ2)e ′ ))f ′ 2 (θ2 + nI2)).
If we fix (θ2, I2) ∈ A, then the first two components of this map are linear with respect
to τ ′ and e ′ , therefore the image of the parallelogram Z + n by the map Fn µ is still a
τ

7.4 - Construction of a symbolic dynamic 171
A 2 n
A 1 n
e
F n µ (Z+ n )
à 1 n à 4 n
à 2 n
à 3 n
A 3 n
A 4 n
Z + n
Figure 5: Position of Z + n and Fn µ (Z+ n )
parallelogram (in a different section), with vertices Ãi n = F n µ(A i n) for i = 1, 2, 3, 4 which
can be explicitly computed.
Similarly, for (τ ′ , e ′ , θ2, I2) ∈ Z − n we compute
F −n
µ (τ ′ , e ′ , θ2, I2) = ((1 + (f2( ¯ θ2)) −1 f2(θ2))τ ′ + e ′ − k + T(−µf2(θ2)τ ′ ),
(f2( ¯ θ2)) −1 f2(θ2)τ ′ ,
θ2 + nI2 − nµf1(τ ′ + e ′ )f ′ 2 (θ2),
I2 + µf1(τ ′ + e ′ )f ′ 2 (θ2)),
so the image of Z − n by the map F −n
µ is a parallelogram with vertices ˜ B i n = F −n
µ (B i n) for
i = 1, 2, 3, 4.
Step 6. Relative position of Z ± n
and F ±n
µ (Z± n ).
Here we will prove that the figures (5) and (6) make sense, that is we will show that
the horizontal (resp. vertical) edges of Fn µ (Z+ −n
n ) (resp. Fµ (Z− n )) are not contained in
Z. The upper horizontal edge of Z + n is the segment joining A2n to A3n , therefore the
upper horizontal edge of Fn µ (Z+ n ) is the segment joining Ã2n to Ã3n . We can compute
e ′ ( Ã2
1 1
n) =
+ µ
f2(θ2 + nI2) f2(θ2)
−1 en − τ ′ 0 + n − T(en + δ e n)
and
e ′ ( Ã3
n ) =
1
f2(θ2 + nI2)
1
+ µ
f2(θ2)
−1 en + τ ′ 0 + n − T(en + δ e n ).
From (63) we have T(en + δ e n ) ≤ n − c3µ 2π−1 , and as τ ′ 0 = c1µ 2π−1 and c3 > c1 + c2 this
gives
We also have
e ′ ( Ã2 n ) > −τ ′ 0 + n − T(en + δ e n )
> (c3 − c1)µ 2π−1 > c2µ 2π−1 > e ′ 0 .
e ′ ( Ã3 n ) = e′ ( Ã2 n ) + 2τ ′ 0 > e′ 0 ,
τ

172 Optimal time of instability for a priori unstable Hamiltonian systems
˜B 4 n
˜B 1 n
Z− n
B 4 n
e
B 3 n
Figure 6: Position of Z − n
B 1 n B 2 n F −n
µ (Z− n )
˜B 3 n
˜B 2 n
and F −n
µ (Z− n )
and so the upper horizontal edge of Z + n is not contained in Z.
For the lower horizontal edge of Fn µ (Z+ n ), which is the segment joining Ã4n to Ã1 n , one
has to prove that e ′ ( Ã4n ) < 0 and e′ ( Ã1n ) < 0. We compute
e ′ ( Ã4
1 1
n ) =
+ µ
f2(θ2 + nI2) f2(θ2)
−1 en + τ ′ 0 + n − T(en − δ e n )
≤ 2πµ −1 en + (c1 − 2c3)µ 2π−1
≤ (c1 − 2c3 + 2π)µ 2π−1 ,
and as 2c3 > c3 > c1 + 2π, this gives e ′ ( Ã4 n) < 0. We also have
e ′ ( Ã1 n ) = e′ ( Ã4 n ) − 2τ ′ 0
< 0,
and so the lower horizontal edge of Z + n is not contained in Z.
Similarly, one can check that the vertical edges of F −n
µ (Z− n ) are not contained in Z,
and this follows from the choice of the constant c4.
set
Step 7. Definition of the maps h ± n .
Our maps h + n and h− n will be suitable extensions of the maps Fn µ
(h + n) |Z + n = (F n µ) |Z + n , (h − n) |Z − n = (F −n
µ ) |Z − n ,
and we want to define Lipschitz extensions of h ± n
Z ∩ (h ± n )−1 (Z) ⊆ Z ± n .
to Z in order to have
τ
−n and Fµ . First we
Let us begin with the maps h + n. Take z = (τ ′ , e ′ , θ2, I2) ∈ Z \ Z + n , then we can find
a unique ˜z = (τ ′ , e ′ (˜z), θ2, I2) ∈ Z + n : indeed, either
en + δ e n
µf2(θ2) < e′ < e ′ 0

7.4 - Construction of a symbolic dynamic 173
in which case we choose
or
and then we choose
Then we set
for some positive constant α + n
is a well-defined Lipschitz extension of F n µ
check that Z ∩ (h + n) −1 (Z) ⊆ Z + n .
e ′ (˜z) = en + δ e n
µf2(θ2) ,
0 < e ′ < en − δ e n
µf2(θ2)
e ′ (˜z) = en − δ e n
µf2(θ2) .
h + n (z) = Fn µ (˜z) + (0, α+ n (e′ (z) − e ′ (˜z)), 0, 0),
yet to be chosen. Hence the map
h + n : Z −→ [−τ ′ 0 , τ ′ 0 ] × F + × A
and, using the form of the extension, one can
For the maps h− n , this is completely analogous. For z = (τ ′ , e ′ , θ2, I2) ∈ Z \ Z− n
we can find a unique ˜z = (τ ′ (˜z), e ′ , θ2, I2) ∈ Z− n where
if
or
if
Then we define
for some constant positive α− n . The map
τ ′ (˜z) = − en
µf2(θ2) + δτ n
− en
µf2(θ2) + δτ n < τ ′ < τ ′ 0,
τ ′ (˜z) = − en
µf2(θ2) − δτ n
−τ ′ 0 < τ ′ < − en
µf2(θ2) + δτ n .
h − n(z) = F −n
µ (˜z) + (α − n(τ ′ (z) − τ ′ (˜z)), 0, 0, 0)
h + n : Z −→ F − × [0, e ′ 0] × A
is a well-defined Lipschitz extension of F −n
µ , with Z ∩ (h − n) −1 (Z) ⊆ Z − n .
Step 8. Verification of Hypotheses (H) and (L’).
, then
Now let us show that the maps h + n and h − n satisfy hypotheses (H) and (L’). Recall
that
X = [−τ ′ 0, τ ′ 0] ⊆ F − = R, Y = [0, e ′ 0] ⊆ F + = R,
and let us write h + n = (f+ n , g+ n ) where
f + n
: X × Y × A −→ X × A

174 Optimal time of instability for a priori unstable Hamiltonian systems
and
For x ∈ X × A, consider the partial map
g + n : X × Y × A −→ F + .
g + n,x : Y −→ F + .
By step 6 (see the figure 5), the image under (h + n ) |Z + n = (Fn µ ) |Z + n
of Z + n do not belong to Z, and this implies that
so the hypothesis (H2) is satisfied.
Y = [0, e ′ 0 ] ⊆ g+ n,x (Y ∩ Z+ n ) ⊆ g+ n,x (Y ),
of the horizontal edges
Now it remains to show hypothesis (H1) and (L’). This follows from the choice of α + n
and lengthy calculations of the various partial derivatives of Fn µ , using both the explicit
expression obtained in step 5 and the estimates of Appendix 7.A. We find the following:
if we define
α + n = π−1 (µ|T ′ n |),
then (H1) is satisfied with
and we estimate the Lipschitz constants
ρ + n = O (µ|T ′ n |) ,
ν + n = O µ −1 |T ′ n |−1 , σ + n = O µ 2π−1 , κ + n = O µ 2π−1 |T ′ n | ,
so (L’) is satisfied for µ small enough as
σ + n + ν+ n max{1, κ+ n } = σ+ n + ν+ n κ+ n = O µ 2π−1 .
The situation for h − n is of course similar.
Step 9. Conclusions.
All the hypotheses of Proposition 7.4 are satisfied, so we can apply it to any sequence
¯n ∈ A: there exist a set Λ ⊆ Z, invariant by h + , and a homeomorphism
n0
Υ : ΣA × A −→ Λ
(¯n, (θ, I)) ↦−→ (Φ¯n(θ, I), (θ, I))
which conjugates h +
n0|Λ to the skew-product on A given by
F¯n(θ, I) = F + n0 (Φ¯n(θ, I), θ, I), (θ, I) ∈ A,
where Φ¯n : A → X × Y is a contracting map, with
Lip(Φ¯n) ≤ sup{(σ
n∈A
+ n (1 − ν+ n κ+ n )−1 ), (σ − n (1 − ν− n κ−n )−1 )} = O µ 2π−1 . (65)
Recall also from (57) that
So we let
h + n0 (Φ¯n(θ, I), θ, I) = (Φ¯n(F¯n(θ, I)), F¯n(θ, I)), (θ, I) ∈ A. (66)
Oµ = Θ −1 (Z), Λµ = Θ −1 (Λ) ⊆ Oµ, Υµ = Θ −1 ◦ Υ.

7.4 - Construction of a symbolic dynamic 175
For x ∈ Λµ,
h +
n x 0 (x) = Fnx 0
µ (x) = ˜ Fµ(x),
where ˜ Fµ is the transversal map of Fµ associated to Oµ. Then by definition of Λµ, the
sequence ¯n x is a well-defined element of ΣA. Setting x = (τx, ex, θx, Ix) ∈ Λµ, the above
conjugacy gives
˜Fµ ◦ Υµ(¯n x , θx, Ix) = Υµ(σ(¯n x ), F¯n x(θx, Ix)).
Now it remains to explain the form of F¯n and the estimates (56). We can write
as
Υµ : ΣA × A −→ Λµ
Υµ(¯n, (θ, I)) = (ψ¯n(θ, I), (θ, I)) = (τ¯n(θ, I), e¯n(θ, I), θ, I),
where τ¯n : A → R and e¯n : A → R are easily deduced from
Φ¯n = (Φ 1 ¯n, Φ 2 ¯n) : A −→ X × Y
and from Θ −1 . Now by definition of X and Y ,
and this gives
|Φ 1 ¯n|C 0 (A) = O µ 2π−1 , |Φ 2 ¯n|C 0 (A) = O µ 2π−1 ,
|τ¯n|C 0 (A) = O µ 2π−1 , |e¯n|C 0 (A) = O µ 2π .
Moreover using (65) these maps are Lipschitz and
Now from (66) we can write
Lip(τ¯n) = O µ 2π−1 , Lip(e¯n) = O µ 2π−1 .
F¯n(θ, I) = (θ + n0I, I + 2µf1(φ¯n(θ, I)) cos 2π(θ + n0I)), ¯n ∈ ΣA, (θ, I) ∈ A,
where the Lipschitz function φ¯n : A → R satisfies the equation
The estimate
φ¯n(θ, I) = τσ(¯n)(θ + n0I, I + 2µf1(φ¯n(θ, I)) cos 2π(θ + n0I)). (67)
|φ¯n| C 0 (A) = O µ 2π−1
is obvious, but for the Lipschitz constant this is more subtle. Let us define
g¯n(θ, I) = (θ + n0I, I + 2µf1(φ¯n(θ, I)) cos 2π(θ + n0I)).
Then on the one hand, using the definition of f1 we can estimate
but on the other hand, from (67)
so
and therefore
This concludes the proof.
Lip(g¯n) ≤ sup{1 + µ −1
2 ln µ −1 , Lip(φ¯n)},
Lip(φ¯n) ≤ Lip(τ¯n)Lip(g¯n) < Lip(g¯n),
Lip(g¯n) = O µ −1
2 ln µ −1
Lip(φ¯n) = O µ 2π−1 2 ln µ −1
.

176 Optimal time of instability for a priori unstable Hamiltonian systems
7.5 Construction of a pseudo-orbit
7.5.0.1. In this section, we will restrict our study to a special type of skew-product
over a Bernoulli shift, which is called "polysystem" (see [Mar08], it is also known as
Iterated Function System).
Recall from definition 7.4 that a polysystem is a skew-product (over σ) such that
for any ¯n = (nk)k∈Z ∈ ΣA, one has F¯n = fn0. So a polysystem does not depend on the
whole sequence ¯n ∈ ΣA but only on its first component n0 ∈ A, hence instead of being
defined by a family of maps indexed by ΣA, it is defined by a family of maps indexed by
A. Then one can easily see that a sequence (nk, xk)k∈Z ∈ A × M gives rise to an orbit
(σ k (¯n), xk)k∈Z ∈ ΣA × M, where ¯n = (nk)k∈Z, for the polysystem [[fn]]n∈A if and only if
fnk (xk) = xk+1, k ∈ Z,
and its projection onto M corresponds to the iteration of the maps (fn)n∈A in the order
prescribed by the sequence ¯n.
7.5.0.2. In the previous section, we showed that the dynamics of the first return map
of our diffeomorphism Fµ in a neighbourhood of the homoclinic annulus is conjugated
to the skew-product map
F¯n(θ, I) = (θ + n0I, I + 2µf1(φ¯n(θ, I)) cos2π(θ + n0I)), ¯n ∈ ΣA, (θ, I) ∈ A.
This family of maps depend on the whole sequence ¯n ∈ ΣA, but as φ¯n is small,
f1(φ¯n(θ, I)) is close to one, and so F¯n is close to
fn0(θ, I) = (θ + n0I, I + 2µ cos2π(θ + n0I)), n0 ∈ A, (θ, I) ∈ A.
These "standard" maps can be seen as perturbations of iterates of the integrable twist
map T(θ, I) = (θ + I, I): more precisely we can write fn = V ◦ T n where
V (θ, I) = (θ, I + 2µ cos2πθ)
is a "vertical" map which is close to identity if µ is small.
We will first construct an orbit for the polysystem [[fn]]n∈A defined by the maps fn,
and in the next section, we will use it as a pseudo-orbit for the skew-product [[F¯n]]¯n∈ΣA
defined by the maps F¯n.
7.5.0.3. The goal of this section is to prove the following proposition.
Proposition 7.5. There exists a positive constant C such that for µ small enough, there
exists an orbit (σ k (¯n), xk)k∈Z ∈ ΣA × A for the polysystem [[fn]]n∈A defined by
such that
and the estimates
hold true.
fn(θ, I) = (θ + nI, I + 2µ cos2π(θ + nI)), n ∈ A, (θ, I) ∈ A,
|IN − I0| ≥ 2,
lim
k→±∞ Ik = ±∞,
N
k=0
nk ≤ Cµ −1 ln µ −1 ,

7.5 - Construction of a pseudo-orbit 177
0 K
−K
Figure 7: Interval Jδ (projection of BK onto T)
We will see that this proposition holds only if we can choose the integers nk, k ∈ Z,
as large as µ −1
2 ln µ −1 , so this explains the choice of
A = {[ln µ −1 ], . . ., 2[µ −1
2 ln µ −1 ]}.
The upper bound on the sum of the integers nk needed to produce a drift of order one
is basically the "time of diffusion" in this context.
In the sequel, we shall only explain the construction of the positive sequence
(nk, θk, Ik)k≥0, since, of course, the construction of the negative sequence (nk, θk, Ik)k≤−1
is completely similar.
7.5.0.4. Let us now describe the construction of our orbit. In all this section we will
need to introduce two real numbers 0 < K < K ′ < √ 2/2. First, we fix K such that
0 < K < √ 2/2 and we define the non-empty domain
BK = {(θ, I) ∈ A | cos 2πθ ≥ K, sin 2πθ ≤ −K}.
We shall only need the first condition cos 2πθ ≥ K in this section, the other condition
sin 2πθ ≤ −K will be used in the next section.
One can also write
BK = {(θ, I) ∈ A | − arccosK
2π
T
Jδ
arcsin K
≤ θ ≤ −
2π
[Z]},
that is BK = Jδ × R (see figure 7), where Jδ is an interval of T of length
δ = arccosK
2π
−
arcsin K
.
2π
Now K being fixed, we shall also consider K ′ ∈ ]K, √ 2/2[ such that the domain BK ′ =
Jδ/2 × R, where Jδ/2 is an interval of length δ/2. Now recall that we have fn = V ◦ T n
with
V (θ, I) = (θ, I + 2µ cos2πθ), T(θ, I) = (θ + I, I), (θ, I) ∈ A
and by definition, the map V leaves the set BK invariant and produce in BK a drift in the
I-direction which is at least equal to 2µK by the condition cos 2πθ ≥ K. Therefore, to

178 Optimal time of instability for a priori unstable Hamiltonian systems
prove the first part of our proposition, we will construct a sequence of points (θk, Ik)k∈Z ∈
A for which we can find a sequence of integers (nk)k∈Z ∈ A such that
T nk (θk, Ik) ∈ BK. (68)
Indeed, in this case (θk+1, Ik+1) = fnk (θk, Ik) = V ◦ T nk(θk, Ik) satisfies
Ik+1 − Ik ≥ 2µK.
Then to prove our second part, we shall need estimates on these integers nk, k ∈ Z. In
fact the relation (68) can be written as
R nk
Ik (θk) = θk + nkIk ∈ Jδ,
where RI is the rotation on the circle T of angle I. Most of the time, that is if either
Ik is irrational or if Ik = p/q with |q| ≥ δ −1 , this will be realized, and the integers nk,
k ∈ Z, can be estimated. This is an easy consequence of the "ergodization" theorem
recalled below, that we shall use crucially in our construction.
7.5.0.5. For I ∈ R, consider the rotation RI(θ) = θ + I defined on the circle T. Given
0 < δ < 1, let Jδ be the set of intervals of T of length δ. We define the δ-ergodization
time N(I, δ) ≤ +∞ by
or equivalently
N(I, δ) = inf{n ∈ N | {θ, . . .,R n I (θ)} ∩ J = ∅, ∀θ ∈ T, ∀J ∈ Jδ},
N(I, δ) = inf{n ∈ N |
0≤k≤n
R −k
I (J) = T, ∀J ∈ Jδ}.
One can easily see that N(I, δ) < +∞ except if I is a rational number with a denominator
smaller than δ −1 , but it is more difficult to prove that when it is defined, this
number is essentially given by the inverse of the distance to these "bad" rationals. This
is the content of the theorem below, which is due to Berti, Biasco and Bolle ([BBB03]).
Theorem 7.6 (Berti-Bolle-Biasco). There exist a positive constant M such that if
then for I ∈ R \ Rδ,
Rδ = {p/q ∈ Q | |q| ≤ Mδ −1 },
N(I, δ/2) ≤ d(I, Rδ) −1 .
This is a consequence of Theorem 4.2 in [BBB03] (see also the estimate (5.3) in
[BBB03]), where the above proposition is proved both for the continuous and multidimensional
case. Of course one can give an explicit value for the numerical constant
M but this will not be useful.
In fact, most of the time we shall use this result in the following form.
Lemma 7.7. Let I ∈ R \ Rδ, θ ∈ T and J ⊆ T any interval of length δ/2. Then for
any m ∈ N, one can find an integer
such that θ + nI ∈ J.
n ∈ [m, m + d(I, Rδ) −1 ]

7.5 - Construction of a pseudo-orbit 179
7.5.0.6. In order to construct our orbit, now we know that we have to consider the
distance to this set of "bad rationals" Rδ.
So given µ > 0, we first define the domain of "fast drift"
DF(µ) = (θ, I) ∈ A | d(I, Rδ) ≥ 1
ln µ −1
,
the domain of "slow drift"
DS(µ) =
and finally the domain of "resonances"
Obviously we have
DR(µ) =
(θ, I) ∈ A | µ1 2
ln µ −1 < d(I, Rδ) < 1
ln µ −1
(θ, I) ∈ A | d(I, Rδ) ≤ µ1 2
ln µ −1
A = DF(µ) ∪ DS(µ) ∪ DR(µ).
But in the sequel, it will be more convenient to further decompose these sets. Indeed,
for p/q ∈ Rδ, if we define
and
DS(µ, p/q) =
DR(µ, p/q) =
(θ, I) ∈ A | µ1 2
1
< |I − p/q| <
ln µ −1 ln µ −1
(θ, I) ∈ A | |I − p/q| ≤ µ1 2
ln µ −1
then, for µ small enough, one has
DS(µ) =
DS(µ, p/q), DR(µ) =
and
p/q∈Rδ
A = DF(µ)
⎛
⎝
p/q∈Rδ
p/q∈Rδ
⎞
DS(µ, p/q) ⎠
⎛
⎝
p/q∈Rδ
.
,
,
DR(µ, p/q)
⎞
DR(µ, p/q) ⎠. (69)
From now on we will assume that µ is sufficiently small, with respect to δ and M which
are fixed, so the above decomposition (69) is in fact a partition of A (since the set Rδ is
discrete) and moreover the following properties hold true: for any (θ, I) ∈ DS(µ), there
exist a unique p/q ∈ Rδ such that (θ, I) ∈ DS(µ, p/q), and if
m = inf{n ∈ N ∗ | fn(θ, I) /∈ DS(µ, p/q)},
then m is well defined and necessarily the point (θm, Im) = fm(θ, I) does not belong
to DS(µ) (either it is in DR(µ) or DF(µ), since the size of the jump, which is of order
µ, is much smaller than the width of the connected components of DR(µ) or DF(µ)).
We also ask the same for a point (θ, I) ∈ DR(µ): under the iteration of a map fn the

180 Optimal time of instability for a priori unstable Hamiltonian systems
p2
q2
p1
q1
DS(µ)
DF(µ)
DR(µ)
Figure 8: Domains DF(µ), DS(µ) and DR(µ)
first time it escapes the domain DR(µ, p/q), it also escapes the domain DR(µ) (in fact
it enters into the domain DS(µ)). This situation is depicted in figure 8.
7.5.0.7. The construction of our orbit will be inductive, and we will start with a
point (θ0, I0) ∈ DF(µ). Then we have the following easy application of the ergodization
theorem.
Lemma 7.8. Let (θ, I) ∈ DF(µ). There exists an integer
such that T n (θ, I) ∈ BK ′.
n ∈ {[ln µ −1 ], . . .,2[lnµ −1 ]}
Proof. Recall that T n (θ, I) ∈ BK ′ if and only if θ + nI ∈ Jδ/2. Now by definition of
DF(µ),
d(I, Rδ) ≥ 1
ln µ −1
so the conclusion follows from Lemma 7.7 by choosing m = [ln µ −1 ].
7.5.0.8. As long as the orbit stays in DF(µ), we can use the previous lemma. Then
it will enter into the domain of slow drift, and this is where the ergodization theorem
gives integers as large as µ −1
2 ln µ −1 . However in the lemma below we will see how, after
iterating a finite number of maps fnk , nk ∈ A, with nk of order µ −1 ln µ −1 , one can
actually cross through this domain of slow drift.
Lemma 7.9. Let (θ, I) ∈ DS(µ). There exist an index j ∈ N and integers n0, . . .,nj
satisfying:
(i) nk ∈ A, for 0 ≤ k ≤ j;
(ii) fnk ◦ · · · ◦ fn0(θ, I) ∈ BK ′, for 0 ≤ k ≤ j;
(iii) fnj ◦ · · · ◦ fn0(θ, I) /∈ DS(µ).
Moreover, if j ∈ N ∗ , then

7.5 - Construction of a pseudo-orbit 181
(iv) j
k=0 nk ≤ (j + 1) lnµ −1 + (2K ′ ) −1 µ −1 ln µ −1 .
Let us point out that in the proof of Proposition 7.5 below, the above lemma will
always be used with j ≥ 1 so item (iv) will be available.
Proof. There exists a unique p/q ∈ Rδ such that (θ, I) ∈ DS(µ, p/q). By Lemma 7.7
(with m = [ln µ −1 ]) and since
d(I, Rδ) = |I − p/q| −1 ≤ µ −1
2 lnµ −1 ,
we can find an integer n0 ∈ A such that fn0(θ, I) ∈ BK ′. Now if fn0(θ, I) /∈ DS(µ), then
we can take j = 0 in the statement and assertions (i), (ii) and (iii) are proven.
Otherwise, we construct inductively (nk)1≤k≤j where j ∈ N is defined by
j = inf{k ∈ N ∗ | fnk ◦ · · · ◦ fn0(θ, I) /∈ DS(µ, p/q)}.
Since our orbit always stays in B ′ K , then j is obviously well-defined, and at each step
we have used Lemma 7.7 so conditions (i) and (ii) are satisfied. Moreover by a previous
remark we can also write
so
and condition (iii) is also satisfied.
j = inf{k ∈ N ∗ | fnk ◦ · · · ◦ fn0(θ, I) /∈ DS(µ)},
fnj ◦ · · · ◦ fn0(θ, I) /∈ DS(µ),
Finally, let us write (θ0, I0) = (θ, I) and (θk+1, Ik+1) = fnk ◦ · · · ◦ fn0(θ0, I0), for
0 ≤ k ≤ j. Since these points belong to BK ′ we have Ik+1 − Ik ≥ 2µK ′ for 0 ≤ k ≤ j
and hence
j j
nk ≤ ln µ −1 j 1
+
|Ik − p/q|
k=0
k=0
k=0
≤ (j + 1) lnµ −1 + (2K ′ ) −1 µ −1
j
k=0
Ik+1 − Ik
|Ik − p/q| .
The second term on the right-hand side is a Riemann sum, hence for µ small enough it
can be estimated by an integral, namely
and as
j
k=0
this eventually gives
Ik+1 − Ik
≤ 2
|Ik − p/q|
j
k=0
1
ln µ
2
−1
µ 1 2
ln µ −1
µ 1 2
1
ln µ −1 ≤|I−p/q|≤
ln µ −1
dI
I
dI
= 2
|I − p/q|
= 2 lnµ−1 2 ≤ ln µ −1 ,
nk ≤ (j + 1) lnµ −1 + (2K ′ ) −1 µ −1 ln µ −1 .
This is exactly (iv), and so this ends the proof.
1
ln µ −1
µ 1 2
ln µ −1
dI
I ,

182 Optimal time of instability for a priori unstable Hamiltonian systems
7.5.0.9. Now that we have escaped the domain of slow drift, we are in the resonant
domain. Here we cannot use any ergodization result. However our point belongs B ′ K ,
and in the lemma below we will show that by iterating j times a map fn, with n of order
ln µ −1 , it can cross the resonant domain while staying in the larger domain BK.
Lemma 7.10. Let (θ, I) ∈ DR(µ) ∩ BK ′. If µ is small enough, there exist integers
and j ∈ N ∗ such that:
n ∈ {[ln µ −1 ], . . .,2[lnµ −1 ]}
(i) f k n(θ, I) ∈ DR(µ) ∩ BK, for 0 ≤ k ≤ j − 1;
(ii) fj n (θ, I) /∈ DR(µ).
Note that it is possible that the point fj n (θ, I) does not belong to BK either, but
then we are back in the domain of slow drift and we can find an integer n ′ ∈ A such
that T n′ (θ, I)) ∈ BK ′ ⊆ BK.
(f j n
Proof. There exists a unique p/q ∈ Rδ such that (θ, I) ∈ DR(µ, p/q) ∩ BK ′. We choose
µ small enough so
Mδ −1 ≤ ln µ −1 ,
and as q ≤ Mδ −1 , we can find a integer d such that n = dq satisfies
n ∈ {[lnµ −1 ], . . .,2[ln µ −1 ]}.
We will also add a further smallness condition on µ by requiring that
2 1
<
K ln µ −1 2π max{arccosK − arccosK′ , arcsin K ′ − arcsin K}.
Let us define inductively
f k n(θ, I) = (θk, Ik), 0 ≤ k ≤ j,
where j is defined as follows: j = inf{j1, j2} with
and
j1 = inf{k ∈ N ∗ | (θk, Ik) /∈ DR(µ, p/q)}
j2 =
Kµ 1
2 ln µ −1
−1
+ 1.
It follows from the definition of j that (θk, Ik) ∈ DR(µ), for 0 ≤ k ≤ j − 1, so now let
us show that (θk, Ik) ∈ BK.
In fact we will prove below by induction on k that
− arccosK′
2π
−
Since n ≤ 2 lnµ −1 this implies
1
knµ 2
ln µ −1 ≤ θk
arcsin K′
≤ − +
2π
knµ 1
arccos K′
− − 2kµ
2π
1 arcsin K′
2 ≤ θk ≤ − + 2kµ
2π
1
2,
2
ln µ −1, 0 ≤ k ≤ j − 1. (70)

7.5 - Construction of a pseudo-orbit 183
and as k ≤ j2, then
arccos K′
− −
2π
2
K ln µ −1 ≤ θk
arcsin K′
≤ − +
2π
arccos K
−
2π
≤ θk ≤ −
arcsin K
.
2π
2
K ln µ −1,
This means that cos 2πθk ≥ K and sin 2πθk ≤ −K, and therefore this gives (θk, Ik) ∈ BK
for 0 ≤ k ≤ j − 1.
So now let us go through the induction, that is let us prove (70). This is obviously
true for k = 0, so let us assume it is satisfied for some 0 ≤ k ≤ j − 2. We have
θk+1 = θk + nrk,
but since Ik ≥ p/q − µ 1
2/ ln µ −1 and n = dq then
θk+1 ≥ θk + dp − nµ 1
2/ lnµ −1 = θk − nµ 1
2/ lnµ −1
as dp is an integer. Then, using the hypothesis of induction this gives
arccos K′
θk+1 ≥ − −
2π
arccos K′
≥ − −
2π
knµ 1
2
ln µ
1
nµ 2
−
−1 ln µ −1
1
(k + 1)nµ 2
ln µ −1
.
Similarly using the fact that Ik ≤ p/q + µ 1
2/ ln µ −1 one obtains
therefore (70) is proven and so is (i).
arcsin K′
θk+1 ≤ − +
2π
(k + 1)nµ 1
2
ln µ −1
,
[Z],
Now assume that j = j2, then since (θk, Ik) ∈ BK for 0 ≤ k ≤ j − 1,
Ij2 ≥ I0 + 2j2Kµ > I0 + 2µ 1
2/ lnµ −1 > p/q + µ 1
2/ ln µ −1 ,
which means that j2 ≥ j1. This is absurd, therefore j = j1, so (θj, Ij) /∈ DR(µ, p/q) and
this means that (θj, Ij) /∈ DR(µ). This proves (ii).
7.5.0.10. Now we can finally conclude the proof of Proposition 7.5.
Proof of Proposition 7.5. We choose any point (θ0, I0) ∈ DF(µ). Applying successively
Lemma 7.8, Lemma 7.9, Lemma 7.10 and Lemma 7.9 once again, in this precise order,
one obtains a positive sequence (nk, θk, Ik)k∈N ∈ A × A which gives a positive orbit for
our polysystem, that is
fnk (θk, Ik) = (θk+1, Ik+1), k ∈ N.
Note that since we have started in DF(µ), at each time Lemma 7.9 is applied with an
integer j ≥ 1.

184 Optimal time of instability for a priori unstable Hamiltonian systems
Now by construction, T nk(θk, Ik) ∈ BK for any k ∈ N, and since
by definition of BK one obtains
This clearly shows that
(θk+1, Ik+1) = V ◦ T nk (θk, Ik),
Ik+1 ≥ Ik + 2µK.
lim
k→+∞ Ik = +∞,
which proves the first part of the statement.
Then as
if we set N = [(µK) −1 ] + 1,
It remains to estimate the sum of integers
To do that, we will write
and
so
Ik − I0 ≥ 2kµK,
IN − I0 ≥ 2.
S =
N
nk.
k=0
σ1 = {k ∈ [0, N] | (θk, Ik) ∈ DF(µ) ∪ DR(µ)}
σ2 = {k ∈ [0, N] | (θk, Ik) ∈ DS(µ)},
S =
nk +
nk = S1 + S2.
k∈σ1
k∈σ2
For each k ∈ σ1, we know that nk ≤ 2 lnµ −1 and hence
S1 ≤ 2
k∈σ1
ln µ −1 ≤ 2N lnµ −1 ≤ 4K −1 µ −1 ln µ −1 .
To estimate S2, first observe that the set Rδ is discrete, so its intersection with the
compact set {I0 ≤ I ≤ IN} is finite. But the latter set is included in {I0 ≤ I ≤ I0 + 3},
so the constant
M = card{p/q ∈ Rδ | I0 ≤ p/q ≤ I0 + 3},
is independent of µ. Then setting
we obtain
σ2(p/q) = {k ∈ [0, N] | (θk, Ik) ∈ DS(µ, p/q)},
S2 =
nk ≤ M
k∈σ2
k∈σ2(p/q)
nk.

7.6 - Proof of Theorem 7.2 185
Now each σ2(p/q) is not reduced to a point, so by Lemma 7.9
This finally gives
k∈σ2(p/q)
nk ≤ N ln µ −1 + (2K ′ ) −1 µ −1 ln µ −1
≤ 2K −1 µ −1 ln µ −1 + (2K ′ ) −1 µ −1 ln µ −1
≤ 2K −1 + (2K ′ ) −1 µ −1 ln µ −1 .
S ≤ (4K −1 )µ −1 ln µ −1 + M 2K −1 + (2K ′ ) −1 µ −1 ln µ −1
≤ Cµ −1 ln µ −1 ,
with C = 2 sup {4K −1 , M (2K −1 + (2K ′ ) −1 )}, and this proves the proposition.
7.6 Proof of Theorem 7.2
7.6.0.1. In this section we will prove Theorem 7.2, and in view of Proposition 7.2, this
will follow easily from the next result.
Proposition 7.6. There exists a positive constant C such that for µ small enough, there
exists an orbit (σ k (¯n), xk)k∈Z ∈ ΣA × A for the skew-product [[F¯n]]n∈A defined by
such that
F¯n(θ, I) = (θ + n0I, I + 2µf1(φ¯n(θ, I)) cos 2π(θ + n0I)), ¯n ∈ ΣA, (θ, I) ∈ A,
and the estimates
hold true.
|I ′ N − I′ 0 | ≥ 1,
lim
k→±∞ I′ k = ±∞
N
k=0
nk ≤ Cµ −1 ln µ −1 ,
The strategy will be to consider the orbit constructed in Proposition 7.5 as a pseudoorbit
for the skew-product. So in order to find a true orbit nearby, we shall need some
hyperbolicity and this will be described below.
7.6.0.2. Recall that
BK = {(θ, I) ∈ A | cos 2πθ ≥ K, sin 2πθ ≤ −K}.
The condition sin 2πθ ≤ −K will be used here to prove the following lemma.
Lemma 7.11. Let x ∈ A such that T n (x) ∈ BK, for n ∈ A, and ˜x ∈ R 2 a lift of x.
Then the eigenvalues λ± of dfn(˜x) are real and for µ small enough, they satisfy
λ+ > 1 + 2 nπµK > 1, λ− < 1 − 2 nπµK + 4nπµK < 1.
Moreover, if e± ∈ R 2 are eigenvectors associated to λ±, and for v ∈ R 2 , v = v+e++v−e−,
then
1
2 |v| ≤ sup{|v+|, |v−|} ≤ a|v|,
with a = O µ −3 4(ln µ −1 ) 1
2 , and where | . | is the supremum norm on R2 .

186 Optimal time of instability for a priori unstable Hamiltonian systems
Proof. Let us write x = (θ, I) ∈ A and ˜x = ( ˜ θ, I) ∈ R 2 . If we define
s = −2πµ sin 2π(θ + nI),
then s > 2πµK since T n (θ, I) = (θ + nI, I) ∈ BK. As
we have
fn(θ, I) = (θ + nI, I + 2µ cos 2π(θ + nI)), n ∈ A, (θ, I) ∈ A,
dfn( ˜ θ, I) =
Then the eigenvalues are real and given by
Therefore we easily obtain
1 n
∈ M2(R).
2s 1 + 2ns
λ± = 1 + ns ± ns(2 + ns).
λ+ > 1 + √ 2ns > 1 + 2 nπµK,
and then using the equality λ+λ− = 1, for µ small enough one finds
This proves the first part of the statement.
Then if we define
one can easily check that the vectors
λ− < 1 − 2 nπµK + 4nπµK.
α± = n −1 (λ± − 1) = s ± n −1 s(2 + ns),
e± =
1
∈ R 2 ,
are eigenvectors associated to λ±. Since |e+| = |e−| = 1, for v ∈ R 2 if we write
it is trivial that
As for the other inequality, note that
If we define
then one can check that
α±
||v|| = sup{|v+|, |v−|}, v = v+e+ + v−e−,
|v| ≤ |v+| + |v−| ≤ 2||v||.
|v| = sup{|v+ + v−|, |α+v+ + α−v−|}.
r± = α+α− − 1 ± (α+ − α−)
1 − α2 ,
+
|α+v+ + α−v−| ≤ |v+ + v−|
if and only if v− = 0, or (v−) −1v+ ≤ r−, or (v−) −1v+ ≥ r+. Hence
|v+ + v−| if v− = 0, or (v−) −1v+ ≤ r−, or (v−) −1v+ ≥ r+, ,
|v| =
|α+v+ + α−v−| if (v−) −1 v+ ≥ r−, or (v−) −1 v+ ≤ r+.

7.6 - Proof of Theorem 7.2 187
If we study all the cases, one finds
Then we can estimate
||v|| ≤ sup 1, |1 + r+| −1 , |1 + r −1
+ | −1 , |1 + r−| −1 , |1 + r −1
− | −1 |v|.
and using the fact that n ∈ A, one finds
with a = O µ −3
4(ln µ −1 ) 1
2
r± = −1 ± √ 2n−1
s + o n−1 µ ,
.
||v|| ≤ a|v|,
7.6.0.3. Let us now describe an abstract fixed point theorem that will be used to find
an orbit close to our pseudo-orbit.
Consider a Banach space (E, | . |) and T : E → E a continuous linear map. Recall
that the spectrum Sp(T) of T is the set of complex numbers λ such that TC − λIdC is
not an automorphism of EC, where EC and TC are the complexifications of E and T.
Given two real numbers κs, κu satisfying 0 < κs < 1 < κu, we say that T is (κs, κu)hyperbolic
if
Sp(T) ∩ {κs ≤ |z| ≤ κu} = ∅.
In such a case, there exists a T-invariant decomposition
and a constant c > 0 such that
E = Es ⊕ Eu, T(Es) ⊆ Es, T(Eu) ⊆ Eu,
|(T|Es) n | ≤ cη n s , |(T|Eu) −n | ≤ cη n u,
for any n ∈ N, ηs < κs, ηu > κu and where | . | is the induced norm on linear operators.
In fact, one can always find a norm . on E which is adapted to T in the following
sense: . is equivalent to | . | and satisfies
(i) xs + xu = sup{xs, xu}, xs ∈ Es, xu ∈ Eu;
(ii) T|Es ≤ κs, (T|Eu) −1 ≤ κ −1
u .
The following theorem is proved in [Yoc95], section 2.1.
Theorem 7.12. Let T : E → E be (κs, κu)-hyperbolic and let U : E → E be a Lipschitz
map such that
ε = Lip(U − T) < ε0 = inf{1 − κs, 1 − κ −1
u }.
Then U has a unique fixed point p ∈ E, and if . is a norm adapted to T,
p < (ε0 − ε) −1 U(0).
7.6.0.4. Now we can prove Proposition 7.6.

188 Optimal time of instability for a priori unstable Hamiltonian systems
Proof of Proposition 7.6. Consider the orbit (σ k (¯n), θk, Ik)k∈Z ∈ ΣA ×A given by Proposition
7.5. Then the proof is an immediate consequence of the following claim: there
exist a sequence (θ ′ k , I′ k )k∈Z ∈ A such that (σ k (¯n), θ ′ k , I′ k )k∈Z ∈ ΣA × A is an orbit for the
skew product and
|Ik − I ′ k| ≤ µ 2 , k ∈ Z.
So let us construct this orbit.
Let xk = (θk, Ik)k∈Z ∈ A, and ˜xk = ( ˜ θk, Ik)k∈Z one of its lift in R 2 . First we define a
linear map
T : (R 2 ) Z −→ (R 2 ) Z
v ↦−→ T(v)
by
(T(v))k = dfnk−1 (˜xk−1).vk−1, k ∈ Z.
Using the supremum norm on R 2 let us define
E = {v = (vk)k∈Z ∈ (R 2 ) Z | sup |vk| < ∞}.
k∈Z
Then E is obviously a Banach space with the norm
|v| = sup |vk|, v ∈ E.
k∈Z
Now recall that for any ˜x = ( ˜ θ, I) ∈ R2 , if s = −2πµ sin2π( ˜ θ + nI), then
1 n
dfn(˜x) =
∈ M2(R).
2s 1 + 2ns
Since the norm on M2(R) induced by the supremum norm on R2 is given by the maximum
of the sums of the absolute values of the elements in each row, we obtain
sup |dfnk
k∈Z
| C0 (R2
) = sup{1
+ nk} = 1 + µ
k∈Z
−1
2 ln µ −1 < ∞,
then T(F) ⊆ F and therefore T is continuous. Moreover, by construction the sequence
(xk)k∈Z satisfies T nk(xk) ∈ BK, so we can apply Lemma 7.11 and each map
Tk−1 : vk−1 ↦−→ dfnk−1 (˜xk−1).vk−1
is a (κ k−1
s , κ k−1
u )-hyperbolic linear map of R 2 , with
and
κ k s = 1 − 2 nkπµK + 4nkπµK < 1, k ∈ Z,
κ k u = 1 + 2 nkπµK > 1, k ∈ Z.
This implies that T is (κs, κu)-hyperbolic, with
Since nk ∈ A, one finds
κs = sup{κ
k∈Z
k s }, κu = inf
k∈Z {κku }.
κs = 1 − 2 √ πK(ln µ −1 ) 1
2µ 1
4 + 4πK ln µ −1 µ 1
2 < 1,

7.6 - Proof of Theorem 7.2 189
and
Let us set
Now if
κu = 1 + 2 √ πK(ln µ −1 ) 1
2µ 1
2 > 1.
ε0 = inf{1 − κs, 1 − κ −1
u } = O µ 1
v = sup vkk,
k∈Z
2(ln µ −1 ) 1
2
where . k is a norm in R 2 adapted to Tk, k ∈ Z, then . is adapted to T, and from
Lemma 7.11
with a = O µ −3
by
Next we define
4(ln µ −1 ) 1
2
.
Note that if v is a fixed point of U, then
1
|v| ≤ v ≤ a|v|, v ∈ E,
2
U : (R 2 ) Z −→ (R 2 ) Z
v ↦−→ U(v)
.
(U(v))k = F σ k−1 (¯n)(˜xk−1 + vk−1) − ˜xk, k ∈ Z.
F σ k−1 (¯n)(˜xk−1 + vk−1) = ˜xk + vk,
so (σ k (¯n), x ′ k )k∈Z ∈ ΣA × A, where x ′ k ∈ A is the projection onto A of ˜x′ k = ˜xk + vk ∈
R 2 , is an orbit for the skew-product. To prove that U has a fixed point, we will use
Theorem 7.12 and for that we need to estimate the Lipschitz constant of ∆ = U − T
with respect to the adapted norm . .
with
First if v, v ′ ∈ E satisfy |v| ≤ µ 2 , |v| ≤ µ 2 , then from Taylor formula one can compute
|∆(v) − ∆(v ′ )| ≤ (L + 2Mµ 2 )|v − v ′ |,
L = sup Lip fnk
k∈Z
− Fσk
2
(¯n) , M = sup |d fnk
k∈Z
|C0 (R2
) .
Using the estimates (56) obtained in Proposition 7.2, one finds
whereas
L = O µ 4π−3/2 lnµ −1 ,
M = O (ln µ −1 ) 2
is obvious from the definition of the maps (fn)n∈A.
However, far away from 0, the maps U and T are not necessarily close, so for v ∈
(R 2 ) Z we define U(v) by
U(v) =
U(v) if |v| ≤ µ 2 ,
U (µ 2 |v| −1 v) + (1 − µ 2 |v| −1 )T(v) if |v| ≥ µ 2 .

190 Optimal time of instability for a priori unstable Hamiltonian systems
Then setting ∆ = U − T, for any v, v ′ ∈ E one easily obtains
and this gives
|∆(v) − ∆(v ′ )| ≤ 2(L + 2Mµ 2 )|v − v ′ |,
∆(v) − ∆(v ′ ) ≤ 4a(L + 2Mµ 2 )v − v ′ .
In particular, this shows U(F) ⊆ F, and the Lipschitz constant of U − T with respect
to the adapted norm . is
ε = Lip(U − T) ≤ 4a(L + 2Mµ 2
) = O µ 5
4(ln µ −1 ) 5
2 .
As
ε = O µ 5
4(lnµ −1 ) 5
2
< ε0 = O µ 1
2(ln µ −1 ) 1
2
we can finally apply Theorem 7.12: U has a unique fixed point v ′ ∈ E such that
and hence
Note that
|U(0)| = sup
k∈Z
v ′ ≤ (ε0 − ε) −1 U(0),
|v ′ | ≤ 2a(ε0 − ε) −1 |U(0)|.
Using the estimates (56) in Proposition 7.2,
{|Fσk (¯n)(˜xk) − ˜xk+1|} = sup{|Fσk
(¯n)(˜xk) − fnk
k∈Z
(˜xk)|}.
sup |Fσk (¯n)(˜xk) − fnk
k∈Z
(˜xk)| = O µ 4π−1
and as 2a(ε0 − ε) −1 = O µ −5/4 , we have in particular
|v ′ | ≤ µ 2 . (71)
By definition of U, this shows that v ′ is in fact a fixed point of U, and therefore the
sequence (σk (¯n), x ′ k )k∈Z ∈ ΣA × A, where x ′ k is the projection onto A of
˜x ′ k = ˜xk + v ′ k = (˜ θ ′ k , I′ k ), k ∈ Z,
is an orbit for the skew-product. If we set x ′ k = (θ′ k , I′ k ) ∈ A, then from (71) we obtain
This concludes the proof.
|Ik − I ′ k| ≤ µ 2 , k ∈ Z.
7.6.0.5. Now we can eventually prove Theorem 7.2.
Proof of Theorem 7.2. Let (σ k (¯n), θ ′ k , I′ k )k∈Z ∈ ΣA ×A be the orbit for the skew-product
obtained in Proposition 7.6. Then, by Proposition 7.2,
and in the original coordinates
Υµ(nk, θ ′ k, I ′ k) = (τk, ek, θ ′ k, I ′ k) ∈ E × A, k ∈ Z,
Ψ −1 (τk, ek, θ ′ k , I′ k ) = (θk 1 , Ik 1 , θk 2 , Ik 2 ) ∈ A2 , k ∈ Z,
is an orbit for the transversal map ˜ Fµ. By definition of the latter one has
F nk
µ (θk 1 , Ik 1 , θk 2 , Ik 2 ) = (θk+1 1 , I k+1
1 , θ k+1
2 , I k+1
2 ), k ∈ Z,
and since I ′ k = I2 k for k ∈ Z, the theorem follows.

7.A - Time-energy coordinates for the pendulum 191
7.A Time-energy coordinates for the pendulum
In this appendix, we recall some elementary facts about the time-energy coordinates for
the simple pendulum. We refer to [MS04], [Mar05] and [LM05] for more details.
7.A.0.1. Consider the simple pendulum defined by the Hamiltonian
and the open domain
We define the energy function
P(θ, I) = 1
2 I2 + cos 2πθ,
E = {(θ, I) ∈ A | 0 < θ < 1, I > 0}.
e(θ, I) = P(θ, I) − 1 = 1
2 I2 + cos 2πθ − 1
and, using {θ = 1/2} as a reference section, the time function
τ(θ, I) =
θ
1
2
dθ
2(e(θ, I) − V (θ))
where V (θ) = cos 2πθ − 1. For positive energy e > 0, the period of motion is given by
T(e) =
1
2
− 1
2
dθ
2(e − V (θ))
and it is a decreasing function. Moreover, we have the equivalent
T(e) ∼0 −(2π) −1 ln e. (72)
For negative energy e < 0, if θ(e) is such that cos 2πθ(e) = 1 + e, then
Therefore if we define
we have a diffeomorphism
T(e) =
θ(e)
−θ(e)
dθ
2(e − V (θ)) .
E∗ = {(τ, e) ∈ R 2 | e > −2, |τ| < 1
2 T(e)}
Ψ : E −→ E∗
(θ, I) ↦−→ (τ, e)
and one can check that it is symplectic. Moreover, in those coordinates (τ, e) the flow
of the pendulum is straightened out, that is
for t small enough.
Φ tP (τ, e) = (τ + t, e).

192 Optimal time of instability for a priori unstable Hamiltonian systems
7.A.0.2. To conclude, for the proof of Proposition 7.2 in section 7.4.2 we shall need
some estimates on the energy and the period of the periodic orbits (of positive energy)
for the pendulum. For n ∈ N ∗ , we let en be the energy of the n-periodic orbit for Φ P ,
that is T(en) = n, we have
en ∼+∞ exp(−2πn). (73)
Then if we define T ′ n = T ′ (en), T ′′
n = T ′′ (en) where T ′ and T ′′ are the first and second
derivatives of the period function, we can deduce from (72) and (73) that
and
T ′ n ∼+∞ −(2π) −1 exp(2πn) (74)
T ′′
n ∼+∞ 2π(T ′ n )2 = (2π) −1 exp(4πn). (75)

8.1 - Introduction 193
8 Time of instability for high-dimensional Hamiltonian
systems
In this chapter, we use a mechanism introduced by Herman, Marco and Sauzin to show
that if a perturbation of a quasi-convex integrable Hamiltonian system is not too small
with respect to the number of degrees of freedom, then the classical exponential stability
estimates do not hold. Indeed, we construct an unstable solution whose drifting time
is polynomial with respect to the inverse of the size of the perturbation. A different
example was already given by Bourgain and Kaloshin, with a linear time of drift but
with a perturbation which is larger than ours. As a consequence, we obtain a better
upper bound on the threshold of validity of exponential stability estimates. This chapter
follows [Bou10a].
8.1 Introduction
8.1.0.1. Consider a near-integrable Hamiltonian system of the form
H(θ, I) = h(I) + f(θ, I)
|f| < ε
with angle-action coordinates (θ, I) ∈ T n × R n , and where f is a small perturbation, of
size ε, in some suitable topology defined by a norm | . |.
If the system is analytic and h satisfies a generic condition, it is a remarkable result
due to Nekhoroshev ([Nek77], [Nek79]) that the action variables are stable for an exponentially
long interval of time with respect to the inverse of the size of the perturbation:
one has
|I(t) − I0| ≤ c1ε b , |t| ≤ c2 exp(c3ε −a ),
for some positive constants c1, c2, c3, a, b and provided that the size of the perturbation
ε is smaller than a threshold ε0. Of course, all these constants strongly depend on the
number of degrees of freedom n, and when the latter goes to infinity, the threshold ε0
and the exponent of stability a go to zero.
More precisely, in the case where h is quasi-convex and the system is analytic or
even Gevrey, then we know that the exponent a is of order n −1 and this is essentially
optimal (see [LN92], [Pös93], [MS02], [LM05] [Zha09] and the sixth chapter for more
information on the optimality of the stability exponent).
8.1.0.2. This fact was used by Bourgain and Kaloshin in [BK05] to show that if the
size of the perturbation is
εn ∼ e −n ,
then there is no exponential stability: they constructed unstable solutions for which the
time of drift is linear with respect to the inverse of the size of the perturbation, that is
|I(τn) − I0| ∼ 1, τn ∼ ε −1
n .
Their motivation was the implementation of stability estimates in the context of Hamiltonian
partial differential equations, which requires to understand the relative dependence
between the size of the perturbation and the number of degrees of freedom. Their

194 Time of instability for high-dimensional Hamiltonian systems
result indicates that for infinite dimensional Hamiltonian systems, Nekhoroshev’s mechanism
does not survive and that "fast diffusion" should prevail. Of course, in their
example, one cannot simply take n = ∞ as the size of the perturbation εn ∼ e −n goes to
zero and the time of instability τn ∼ e n goes to infinity exponentially fast with respect to
n. A more precise interpretation concerns the threshold of validity ε0 in Nekhoroshev’s
theorem: it has to satisfy
ε0

8.2 - Main result 195
Let us recall that, given α ≥ 1 and L > 0, a function f ∈ C∞ (Tn × B) is (α, L)-Gevrey
if, using the standard multi-index notation,
|f|α,L =
L |l|α (l!) −α |∂ l f| C0 (Tn ×B) < ∞.
l∈N 2n
The space of such functions, with the above norm, is a Banach space that we denote by
G α,L (T n × B). One can see that analytic functions correspond exactly to α = 1.
8.2.0.2. Now we can state our theorem.
Theorem 8.1. Let n ≥ 3, R > 1, α > 1 and L > 0. Then there exist positive constants
c, γ, C and n0 ∈ N ∗ depending only on R, α and L such that for any n ≥ n0, the following
holds: there exists a function fn ∈ G α,L (T n × B) with εn = |fn|α,L satisfying
e −2(n−2) ln(4n ln 2n) ≤ εn ≤ c e −2(n−2) ln(n ln 2n) ,
such that the Hamiltonian system Hn = h + fn has an orbit (θ(t), I(t)) for which the
estimates
|I(τn) − I0| ≥ 1,
nγ c
τn ≤ C ,
hold true.
As we have already explained, this statement gives an upper bound on the threshold
of applicability of Nekhoroshev’s estimates, which is an important issue when trying
to use abstract stability results for "realistic" problems, for instance for the so-called
planetary problem (see [Nie96]).
So let us consider the set of Gevrey quasi-convex integrable Hamiltonians H =
H(n, R, α, L, M, m) defined as follows: h ∈ H if h ∈ G α,L (B) and satisfies both
and
∀I ∈ B, |∂ k h(I)| ≤ M, 1 ≤ |k1| + · · · + |kn| ≤ 3,
∀I ∈ B, ∀v ∈ R n , ∇h(I).v = 0 =⇒ ∇ 2 h(I)v.v ≥ m|v| 2 .
From Nekhoroshev’s theorem (see [MS02] for a statement in Gevrey classes), we know
that there exists a positive constant ε0(H) = ε0(n, R, α, L, M, m) such that the following
holds: for any h ∈ H, there exist positive constants c1, c2, c3, a and b such that if
εn
f ∈ G α,L (T n × B), |f|α,L < ε0(H),
then any solution (θ(t), I(t)) of the system H = h + f, with I(0) ∈ BR/2, satisfies
|I(t) − I0| ≤ c1ε b , |t| ≤ c2 exp(c3ε −a ).
Then we can state the following corollary of our Theorem 8.1.
Corollary 8.2. With the previous notations, one has the upper bound
ε0(H) < e −2(n−2) ln(4n ln2n) .

196 Time of instability for high-dimensional Hamiltonian systems
This improves the upper bound ε0(H) < e −n obtained in [BK05]. For concrete
Hamiltonians like the planetary problem, the actual distance to the integrable system
is slightly less than 10 −3 , hence the above corollary yields the impossibility to apply
Nekhoroshev’s estimates for n > 3.
8.2.0.3. Theorem 8.1 is easily obtained from the example in [MS02]: in fact, the
construction is completely similar but one has to pay attention to the dependence with
respect to n of the various constants involved. Such a result can easily be obtained in
the analytic category, using the more elaborated techniques developed in [LM05], but we
will restrict to the Gevrey case in order to describe entirely this very simple mechanism
of instability.
As the reader will see, we will use only rough estimates leading to the factor n in the
time of instability: this can be easily improved but we do not know if it is possible in
our case (that is with a perturbation of size e −nln(n ln n) ) to obtain a linear time of drift.
Finally, as in [MS02], we have restricted the perturbation to a compact subset of
A n = T n × R n just in order to evaluate Gevrey norms. In fact, the Hamiltonian vector
field generated by Hn = h+fn is complete and the unstable solution (θ(t), I(t)) satisfies
lim
t→±∞ I1(t) = ±∞,
which means that it is bi-asymptotic to infinity.
8.2.0.4. In this text, we will have to deal with time-one maps associated to Hamiltonian
flows. So given a function H, we will denote by Φ H t the time-t map of its Hamiltonian
flow and by Φ H = Φ H 1 the time-one map. We shall use the same notation for timedependent
functions H, that is Φ H will be the time-one map of the Hamiltonian isotopy
(the flow between t = 0 and t = 1) generated by H.
8.3 Proof of Theorem 8.1
The proof of Theorem 8.1 is contained in section 8.3.2, but first in section 8.3.1 we recall
the mechanism of instability presented in the paper [MS02] (see also [MS04]).
This mechanism has two main features. The first one is that it deals with perturbation
of integrable maps rather than perturbation of integrable flows, and then the
latter is recovered by a suspension process. This point of view, which is only of technical
matter, was already used for example in [Dou88] and offers more flexibility in the
construction. The second feature, which is the most important one, is that instead of
trying to detect instability in a map close to integrable by means of the usual splitting
estimates, we will start with a map having already "unstable" orbits and try to embed
it in a near-integrable map. This will be realized through a "coupling lemma", which is
really the heart of the mechanism.
As we will see, the construction offers an easy and very efficient way of computing
the drifting time of unstable solutions, therefore avoiding all the technicalities that are
usually required for such a task.

8.3 - Proof of Theorem 8.1 197
8.3.1 The mechanism
•
•
•
•
•
ψq
ψ −1
q
Figure 9: Drifting point for the map ψq
I = q −1
I = 0
I = −q −1
8.3.1.1. Given a potential function U : T → R, we consider the following family of
maps ψq : A → A defined by
ψq(θ, I) = θ + qI, I − q −1 U ′ (θ + qI) , (θ, I) ∈ A,
for q ∈ N ∗ . If we require U ′ (0) = −1, for example if we choose
U(θ) = −(2π) −1 sin 2πθ, U ′ (θ) = − cos(2πθ),
then it is easy to see that ψq(0, 0) = (0, q −1 ) and by induction
ψ k q (0, 0) = (0, kq−1 ) (76)
for any k ∈ Z (see figure 9). After q iterations, the point (0, 0) drifts from the circle I = 0
to the circle I = 1 and it is bi-asymptotic to infinity, in the sense that the sequence
k ψq (0, 0)
is not contained in any semi-infinite annulus of A. Clearly these maps
k∈Z
are exact-symplectic, but obviously they have no invariant circles and so they cannot
be "close to integrable". However, we will use the fact that they can be written as a
composition of time-one maps,
ψq = Φ q−1 U ◦
Φ 1
2I2 ◦ · · · ◦ Φ 1
2I2 = Φ q−1
U
◦ Φ 1
2I2 q
, (77)
to embed ψq in the q th -iterate of a near-integrable map of A n , for n ≥ 2. To do so, we
will use the following "coupling lemma", which is easy but very clever.
Lemma 8.3 (Herman-Marco-Sauzin). Let m, m ′ ≥ 1, F : Am → Am and G : Am′ →
Am′ two maps, and f : Am → R and g : Am′ → R two Hamiltonian functions generating
complete vector fields. Suppose there is a point a ∈ Am′ which is q-periodic for G and
such that the following "synchronisation" conditions hold:
g(a) = 1, dg(a) = 0, g(G k (a)) = 0, dg(G k (a)) = 0, (S)

198 Time of instability for high-dimensional Hamiltonian systems
for 1 ≤ k ≤ q − 1. Then the mapping
is well-defined and for all x ∈ A m ,
Ψ = Φ f⊗g ◦ (F × G) : A m+m′
−→ A m+m′
Ψ q (x, a) = (Φ f ◦ F q (x), a).
The product of functions acting on separate variables was denoted by ⊗, i.e.
f ⊗ g(x, x ′ ) = f(x)g(x ′ ) x ∈ A m , x ′ ∈ A m′
.
Let us give an elementary proof of this lemma since it is a crucial ingredient.
Proof. First note that since the Hamiltonian vector fields Xf and Xg are complete, an
easy calculation shows that for all x ∈ Am , x ′ ∈ Am′ and t ∈ R, one has
Φ f⊗g
t (x, x ′
) =
Φ g(x′ )f
t
(x), Φ f(x)g
t (x ′ )
and therefore Xf⊗g is also complete. Using the above formula and condition (S), the
points (F k (x), G k (a)), for 1 ≤ k ≤ q − 1, are fixed by Φ f⊗g and hence
Since a is q-periodic for G this gives
and we end up with
using once again (S) and (78).
Ψ q−1 (x, a) = (F q−1 (x), G q−1 (a)).
Ψ q (x, a) = Φ f⊗g (F q (x), a),
Ψ q (x, a) = (Φ f (F q (x)), a)
Therefore, if we set m = 1, F = Φ 1
2 I2 1 and f = q −1 U in the coupling lemma, the
q th -iterate Ψ q will leave the submanifold A × {a} invariant, and its restriction to this
annulus will coincide with our "unstable map" ψq. Hence, after q 2 iterations of Ψ, the
I1-component of the point ((0, 0), a) ∈ A 2 will move from 0 to 1.
8.3.1.2. The difficult part is then to find what kind of dynamics we can put on the
second factor to apply this coupling lemma. In order to have a continuous system with
n degrees of freedom at the end, we may already choose m ′ = n − 2 so the coupling
lemma will give us a discrete system with n − 1 degrees of freedom.
First, a natural attempt would be to try
Indeed, in this case
G = Gn = Φ 1
2 I2 2 +···+1
2 I2 n−1.
F × Gn = Φ 1
2 I2 2 +···+1
2 I2 n = Φ h
and the unstable map Ψ given by the coupling lemma appears as a perturbation of
the form Ψ = Φ u ◦ Φ h , with u = f ⊗ g. However, this cannot work. Indeed, for
(78)

8.3 - Proof of Theorem 8.1 199
j ∈ {2, . . ., n − 1}, one can choose a pj-periodic point a (j) ∈ A for the map Φ 1
2 I2 j , and
then setting
an = (a (2) , . . .,a (n−1) ) ∈ A n−2 , qn = p2 · · ·pn−1,
the point an is qn-periodic for Gn provided that the numbers pj are mutually prime.
One can easily see that the latter condition will force the product qn to converge to
infinity when n goes to infinity. So necessarily the point an gets arbitrarily close to its
first iterate Gn(an) when n (and therefore qn) is large: this is because qn-periodic points
for Gn are equi-distributed on qn-periodic tori. As a consequence, a function gn with
the property
gn(an) = 1, gn(Gn(an)) = 0,
will necessarily have very large derivatives at an if qn is large. Then as the size of the
perturbation is essentially given by
|f ⊗ gn| = |q −1
n U ⊗ gn| = |q −1
n ||gn|,
one can check that it is impossible to make this quantity converge to zero when the
number of degrees of freedom n goes to infinity.
8.3.1.3. As in [MS02], the idea to overcome this problem is the following one. We
introduce a new sequence of "large" parameters Nn ∈ N ∗ and in the second factor we
consider a family of suitably rescaled penduli on A given by
Pn(θ2, I2) = 1
2 I2 2
−2
+ Nn V (θ2),
where V (θ) = − cos 2πθ. The other factors remain unchanged, so
Gn = Φ 1
2 (I2 2 +I2 3 +···+I2 −2
n−1 )+Nn V (θ2)
.
In this case, the map Ψ given by the coupling lemma is also a perturbation of Φ h but of
the form Ψ = Φ u ◦Ψ h+v , with v = N −2
n V . But now for the map Gn, due to the presence
of the pendulum factor, it is now possible to find a periodic orbit with an irregular
distribution: more precisely, a qn-periodic point an such that its distance to the rest of
its orbit is of order N −1
n , no matter how large qn is.
8.3.1.4. Let us denote by (pj)j≥0 the ordered sequence of prime numbers and let us
choose Nn as the product of the n − 2 prime numbers {pn+3, . . .,p2n}, that is
Nn = pn+3pn+4 · · ·p2n ∈ N ∗ .
Our goal is to prove the following proposition.
Proposition 8.1. Let n ≥ 3, α > 1 and L1 > 0. Then there exist a function gn ∈
G α,L1 (A n−2 ), a point an ∈ A n−2 and positive constants c1 and c2 depending only on α
and L1 such that if
Mn = 2 c1Nne c2(n−2)p 1
α
2n , qn = NnMn,
then an is qn-periodic for Gn and (gn, Gn, an, qn) satisfy the synchronization conditions
(S):
gn(an) = 1, dgn(an) = 0, gn(G k n (an)) = 0, dgn(G k n (an)) = 0,

200 Time of instability for high-dimensional Hamiltonian systems
− 1
2
b
×
×
×
× ×
× ×
× ×
M
for 1 ≤ k ≤ qn − 1. Moreover, the estimate
holds true.
−σ σ
Figure 10: The point b M and its iterates
q −1
n |gn|α,L1 ≤ N −2
n , (79)
The rest of this section is devoted to the proof of the above proposition. Note that
together with the coupling lemma (Lemma 8.3), this proposition easily gives a result
of instability (analogous to Proposition 2.1 in [MS02]) for a discrete system which is a
perturbation of the map Φ h .
8.3.1.5. We first consider the simple pendulum
P(θ, I) = 1
2 I2 + V (θ), (θ, I) ∈ A.
With our convention, the stable equilibrium is at (0, 0) and the unstable one is at
(0, 1/2). Given any M ∈ N ∗ , there is a unique point b M = (0, IM) which is M-periodic
for Φ P (this is just the intersection between the vertical line {0} × R and the closed
orbit for the pendulum of period M). One can check that IM ∈ ]2, 3[ and as M goes to
infinity, (0, IM) tends to the point (0, 2) which belongs to the upper separatrix. Since
Pn(θ, I) = 1
2 I2 + N −2
n V (θ), then one can see that
Φ Pn = (Sn) −1 −1
Nn P
◦ Φ ◦ Sn,
where Sn(θ, I) = (θ, NnI) is the rescaling by Nn in the action components. Therefore
the point bM −1
n = (0, Nn IM) is qn-periodic for ΦPn , for qn = NnM. Let (ΦP t )t∈R be the
flow of the pendulum, and
Φ P t (0, IM) = (θM(t), IM(t)).
The function θM(t) is analytic. The crucial observation is the following simple property
of the pendulum (see Lemma 2.2 in [MS02] for a proof).
Lemma 8.4. Let σ = −1 2 + 2 π arctaneπ < 1
2 . For any M ∈ N∗ ,
for t ∈ [1/2, M − 1/2].
θM(t) /∈ [−σ, σ],
1
2

8.3 - Proof of Theorem 8.1 201
Hence no matter how large M is, most of the points of the orbit of b M ∈ A will be
outside the set {−σ ≤ θ ≤ σ} × R (see figure 10). The construction of a function that
vanishes, as well as its first derivative, at these points, will be easily arranged by means
of a function, depending only on the angle variables, with support in {−σ ≤ θ ≤ σ}.
As for the other points, it is convenient to introduce the function
τM : [−σ, σ] −→ ] − 1/2, 1/2[
which is the analytic inverse of θM. One can give an explicit formula for this map:
τM(θ) =
θ
0
dϕ
I 2 M − 4 sin 2 πϕ .
In particular, it is analytic and therefore it belongs to G α,L1 ([−σ, σ]) for α ≥ 1 and
L1 > 0, and one can obtain the following estimate (see Lemma 2.3 in [MS02] for a
proof).
Lemma 8.5. For α > 1 and L1 > 0,
Λ = sup
M∈N∗ |τM|α,L1 < +∞.
Note that Λ depends only on α and L1. Under the action of τM, the points of the
orbit of b M whose projection onto T belongs to {−σ ≤ θ ≤ σ} get equi-distributed, and
we can use the following elementary lemma.
Lemma 8.6. For p ∈ N ∗ , the analytic function ηp : T → R defined by
satisfies
for 1 ≤ k ≤ p − 1, and
p−1
1
ηp(θ) = cos 2πlθ
p
l=0
2
ηp(0) = 1, η ′ p(0) = 0, ηp(k/p) = η ′ p(k/p) = 0,
|ηp|α,L1 ≤ e 2αL1(2πp) 1 α .
The proof is trivial (see [MS02], Lemma 2.4).
8.3.1.6. We can now pass to the proof of Proposition 8.1.
Proof of Proposition 8.1. For α > 1 and L1 > 0, consider the bump function ϕα,L1 ∈
G α,L1 (T) given by Lemma 8.8 (see Appendix 8.A).
We choose our function gn ∈ Gα,L1 n−2 (A ), depending only on the angle variables, of
the form
gn = g (2)
n ⊗ · · · ⊗ g(n−1) n ,
where
g (2)
n (θ2) = ηpn+3(τMn(θ2))ϕ −
α,(4σ) 1 ((4σ) α L1
−1 θ2),

202 Time of instability for high-dimensional Hamiltonian systems
and
Let us write
Now we choose our point an = (a (2)
and
g (i)
n (θi) = ηpn+1+i (θi), 3 ≤ i ≤ n − 1.
a (i)
n
c1 = ϕ −
α,(4σ) 1 α L1 −
α,(4σ) 1 α L1
a (2)
n
n , . . .,a (n−1)
n
= bMn
n
) ∈ A n−2 . We set
= (0, N −1
n IMn),
= (0, p−1 n+1+i ), 3 ≤ i ≤ n − 1.
Let us prove that an is qn-periodic for Gn. We can write
Gn = Φ 1
2 I2 −2 1
2 +Nn V (θ2)
× Φ 2 (I2 3 +I2 4 +···+I2 n−1 ) = Φ Pn × G.
Since pn+4, . . .,p2n are mutually prime, the point (a (3)
G, with period
N ′ n = pn+4 · · ·p2n.
.
n , . . .,a (n−1)
n
) ∈ A n−3 is periodic for
By construction, the point a (2)
n = b Mn
n ∈ A is periodic for ΦPn , with period qn = NnMn,
where
Nn = pn+3pn+4 · · ·p2n.
This means that an is periodic for the product map Gn, and the exact period is given
by the least common multiple of qn and N ′ n . Since N ′ n divides qn, the period of an is qn.
Now let us show that the synchronization conditions (S) hold true, that is
gn(an) = 1, dgn(an) = 0, gn(G k n (an)) = 0, dgn(G k n (an)) = 0,
for 1 ≤ k ≤ qn − 1. Since ϕα,L1(0) = 1, then
and as ϕ ′ (0) = 0, then
α,L1
gn(an) = g (2)
n
(0) · · ·g(n−1) n (0) = 1
dgn(an) = 0.
To prove the other conditions, let us write G k n (an) = (θk, Ik) ∈ A n−2 , for 1 ≤ k ≤ qn −1.
If θ (2)
k
does not belong to ] − σ, σ[, then g(2) n and its first derivative vanish at θ (2)
k
, so
because it is the case for ϕ α,(4σ) − 1 α L
Otherwise, if −σ < θ (2)
k
and therefore
gn(θk) = dgn(θk) = 0.
< σ, one can easily check that
− Nn − 1
2
≤ k ≤ Nn − 1
2
τMn(θ (2) k
k ) = ,
Nn

8.3 - Proof of Theorem 8.1 203
while
θ (i)
k
k
= , 3 ≤ i ≤ n − 1.
pn+i+1
If N ′ n = pn+4 · · ·p2n divides k, that is k = k ′ N ′ n for some k ′ ∈ Z, then
τMn(θ (2) k
k ) =
Nn
= k′
pn+3
and therefore, by Lemma 8.6, ηpn+3 vanishes with its differential at θ (2)
k , and so does
g (2)
n . Otherwise, N ′ n
functions ηpn+1+i
does not divide k and then, for 3 ≤ i ≤ n − 1, at least one of the
vanishes with its differential at θ(2)
k
gn(θk) = dgn(θk) = 0, 1 ≤ k ≤ qn − 1,
and the synchronization conditions (S) are satisfied.
, and so does g(i)
n . Hence in any case
Now it remains to estimate the norm of the function gn. First, using Lemma 8.9,
one finds
|gn|α,L1 ≤ ϕ −
α,(4σ) 1 α L1 −
α,(4σ) 1 |ηpn+3 ◦ τMn|α,L1|ηpn+4|α,L1|ηp2n|α,L1,
α L1
which by definition of c1 gives
|gn|α,L1 ≤ c1|ηpn+3 ◦ τMn|α,L1|ηpn+4|α,L1|ηp2n|α,L1.
Then, by definition of Λ (Lemma 8.5) and using Lemma 8.10 (with Λ1 = Λ 1
α),
|ηpn+3 ◦ τMn|α,L1 ≤ |ηpn+3| α,Λ 1 α ,
so using Lemma 8.6 and setting c2 = 2α sup Λ 1
α, L1 (2π) 1
α, this gives
Finally, by definition of Mn we obtain
and as qn = NnMn, we end up with
This concludes the proof.
|gn|α,L1 ≤ c1e 2α(Λ 1 α +(n−3)L1)(2πp2n) 1 α
2α sup Λ
≤ c1e 1
α ,L1 (n−2)(2πp2n) 1 α
≤ c1e c2(n−2)p 1 α 2n.
|gn|α,L1 ≤ MnN −1
n ,
q −1
n |gn|α,L1 ≤ N −2
n .

204 Time of instability for high-dimensional Hamiltonian systems
8.3.2 Proof of Theorem 8.1
8.3.2.1. In the previous section, we were concerned with a perturbation of the integrable
diffeomorphism Φ h , which can be written as Φ u ◦ Φ h+v . So now we will briefly
describe a suspension argument to go from this discrete case to a continuous case (we
refer once again to [MS02] for the details).
Here we will make use of bump functions, however the process is still valid, though
more difficult, in the analytic category, (see for example [Dou88] or [KP94]). The basic
idea is to find a time-dependent Hamiltonian function on A n such that the time-one
map of its isotopy is Φ u ◦ Φ h+v , or, equivalently, an autonomous Hamiltonian function
on A n+1 such that its first return map to some 2n-dimensional Poincaré section coincides
with our map Φ u ◦ Φ h+v .
Given α > 1 and L > 1, let us define the function
−1 φα,L =
ϕα,L
T
ϕα,L,
1
where ϕα,L is the bump function given by Lemma 8.8. If φ0(t) = φα,L
t − and 4
3
φ1(t) = φα,L t − , the time-dependent Hamiltonian
clearly satisfies
4
H ∗ (θ, I, t) = (h(I) + v(θ)) ⊗ φ0(t) + u(θ) ⊗ φ1(t)
Φ H∗
= Φ u ◦ Φ h+v .
But as u and v go to zero, H ∗ converges to h⊗φ0 rather than h. However, using classical
generating functions, it is not difficult to modify the Hamiltonian in order to prove the
following proposition (see Lemma 2.5 in [MS02]).
Lemma 8.7. Let n ≥ 1, R > 1, α > 1, L1 > 0 and L > 0 satisfying
L α 1 = Lα (1 + (L α + R + 1/2)|φα,L|α,L). (80)
If u, v ∈ G α,L1 (T n−1 ), there exists f ∈ G α,L (T n × B), independent of the variable In,
such that if
H(θ, I) = 1
2 (I2 1 + · · · + I 2 n−1) + In + f(θ, I), (θ, I) ∈ A n ,
for any energy e ∈ R, the Poincaré map induced by the Hamiltonian flow of H on the
section {θn = 0} ∩ H −1 (e) coincides with the diffeomorphism
Moreover, one has
Φ u ◦ Φ h+v .
sup{|u| C 0, |v| C 0} ≤ |f|α,L ≤ c3 sup{|u|α,L1, |v|α,L1}, (81)
where c3 = 2|φα,L|α,L depends only on α and L.
8.3.2.2. Now we can finally prove our theorem.

8.3 - Proof of Theorem 8.1 205
Proof of Theorem 8.1. Let R > 1, α > 1 and L > 0, and choose L1 satisfying the
relation (80). The constants c1 and c2 of Proposition 8.1 depend only on α and L1,
hence they depend only on R, α and L.
We can define un, vn ∈ G α,L1 (T n−1 ) by
un = q −1
n U ⊗ gn, vn = N −2
n V,
where U(θ1) = −(2π) −1 sin 2πθ1, V (θ2) = − cos 2πθ2 (so vn is formally defined on T but
we identify it with a function on T n−1 ) and gn is the function given by Proposition 8.1.
Let us apply Lemma 8.7: there exists fn ∈ G α,L (T n × B), independent of the variable
In, such that if
Hn(θ, I) = 1
2 (I2 1 + · · · + I 2 n−1) + In + fn(θ, I), (θ, I) ∈ A n ,
for any energy e ∈ R, the Poincaré map induced by the Hamiltonian flow of H on the
section {θn = 0} ∩ H −1 (e) coincides with the diffeomorphism
Φ un ◦ Φ 1
2 (I2 1 +···+I2 n−1 )+vn = Φ un ◦ Φ h+vn .
Let us show that our system Hn has a drifting orbit. First consider its Poincaré section
defined by
with
Ψn = Φ un ◦ Φ 1
2 (I2 1 +···+I2 n−1 )+vn = Φ fn⊗gn ◦ (F × Gn),
fn = q −1
n U, F = Φ12
I2 1, Gn = Φ 1
2 (I2 2 +I2 3 +···+I2 −2
n−1 )+Nn V (θ2)
.
By Proposition 8.1, we can apply the coupling lemma (Lemma 8.3), so
Then, using (77), observe that
so
Ψ qn
n ((0, 0), an) = (Φ f n ◦ F qn (0, 0), an).
Φ fn ◦ F qn = Φ q−1
n U ◦
Φ 1
2I2 qn 1 = ψqn,
Ψ q2 n
n ((0, 0), an) = ((Φ fn ◦ F qn ) qn (0, 0), an)
= (ψ qn
(0, 0), an)
qn
= ((0, 1), an),
where the last equality follows from (76). Hence, after q2 n iterations, the I1-component
of the point xn = ((0, 0), an) ∈ An−1 drifts from 0 to 1. Then, for the continuous system,
the initial condition (xn, t = 0, In = 0) in An gives rise to a solution (x(t), t, In(t)) =
(x(t), θn(t), In(t)) of the Hamiltonian vector field generated by Hn such that
x(k) = Ψ k n (xn), k ∈ Z.
So after a time τn = q 2 n, the point (xn, (0, 0)) drifts from 0 to 1 in the I1-direction, and
this gives our drifting orbit.
Now let εn = |fn|α,L1 be the size of our perturbation. Using the estimate (79)
and (81) one finds
N −2
. (82)
n ≤ εn ≤ c3N −2
n

206 Time of instability for high-dimensional Hamiltonian systems
By the prime number theorem, pn is equivalent to n lnn, so there exists n0 ∈ N ∗ such
that for n ≥ n0, one can ensure that
which gives
p2n/4 ≤ pn+i ≤ p2n, 3 ≤ i ≤ n,
(p2n/4) n−2 ≤ Nn ≤ p n−2
2n , N 1
n
n−2
≤ p2n ≤ 4N 1
n−2
n . (83)
We can also assume by the prime number theorem that for n ≥ n0, one has
2n ln2n ≤ p2n ≤ 2(2n ln 2n) = 4n ln2n. (84)
From the above estimates (83) and (84) one easily obtains
and, together with (82), one finds
e (n−2) ln(2−1 n ln2n) ≤ Nn ≤ e (n−2) ln(4n ln2n) , (85)
e −2(n−2) ln(4n ln2n) ≤ εn ≤ c3e −2(n−2) ln(2−1 nln 2n) . (86)
Finally it remains to estimate the time τn. First recall that
and with (83)
Hence
Then using (85) we have
and from (82) we know that
so we obtain
Mn = 2
c1Nne c2(n−2)p 1 α 2n
qn = NnMn ≤ 3c1N 2 ne 4c2(n−2)N
q 2 n ≤ 9c2 1 N4 n e8c2(n−2)N
N
q 2 n ≤ 9c2 1
1
α(n−2)
n
N 4 n ≤
c3
εn
,
1
α(n−2)
n .
≤ (4n ln 2n) 1
α
c3
εn
2
,
1
α(n−2)
n .
2
e 8c2(n−2)(4n ln2n) 1 α .
Now taking n0 larger if necessary, as α > 1, one can ensure that for n ≥ n0,
so
Therefore
(4n) 1
α ≤ n, (ln 2n) 1
α ≤ ln(2 −1 n ln 2n),
8c2(n − 2)(4n ln2n) 1
α ≤ 8c2(n − 2)n ln(2 −1 n ln 2n).
q 2 n ≤ 9c 2
c3
1
εn
≤ 9c 2
c3
1
εn
2
e 8c2n(n−2)ln(2 −1 n ln 2n)
2
e 2(n−2) ln(2−1 n ln2n))
4c2n
.

8.A - Gevrey functions 207
Finally by (86) we obtain
q 2 n ≤ 9c2 1
c3
c
≤ C
εn
nγ εn
2 c3
εn
4c2n
with C = 9c 2 1 , c = c3 and γ = 2 + 4c2. This ends the proof.
8.A Gevrey functions
In this very short appendix, we recall some facts about Gevrey functions that we used
in the text. We refer to [MS02], Appendix A, for more details.
The most important property of α-Gevrey functions is the existence, for α > 1, of
bump functions.
Lemma 8.8. Let α > 1 and L > 0. There exists a non-negative 1-periodic function
ϕα,L ∈ Gα,L [−1 1 , 2 2 ] whose support is included in [−1 1 , 4 4 ] and such that ϕα,L(0) = 1 and
(0) = 0.
ϕ ′ α,L
The following estimate on the product of Gevrey functions follows easily from the
Leibniz formula.
Lemma 8.9. Let L > 0, and f, g ∈ G α,L (T n × B). Then
|fg|α,L ≤ |f|α,L|g|α,L.
Finally, estimates on the composition of Gevrey functions are much more difficult
(see Proposition A.1 in [MS02]), but here we shall only need the following statement.
Lemma 8.10. Let α ≥ 1, Λ1 > 0, L1 > 0, and I, J be compact intervals of R. Let
f ∈ G α,Λ1 (I), g ∈ G α,L1 (J) and assume g(J) ⊆ I. If
then f ◦ g ∈ G α,L1 (J) and
|g|α,L1 ≤ Λ α 1 ,
|f ◦ g|α,L1 ≤ |f|α,Λ1.

208 Time of instability for high-dimensional Hamiltonian systems

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