PhD thesis - IAS

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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
Page 1 and 2: N o d’ordre: 9910 THÈSE Présent
Page 3: Résumé Dans cette thèse, on s’
Page 7: Béton style ... Quelque soit l’
Page 11: Remerciements À mes deux excellent
Page 15: Première partie Introduction, rés
Page 18 and 19: 18 Introduction En réalité, ces
Page 20 and 21: 20 Introduction dégénérée. Il n
Page 22 and 23: 22 Introduction (ensembles, fonctio
Page 24 and 25: 24 Introduction (i) N est un tore p
Page 26 and 27: 26 Introduction 1.2.2 Dynamique des
Page 28 and 29: 28 Introduction D’un autre point
Page 30 and 31: 30 Introduction 1.3.2.4. Détaillon
Page 32 and 33: 32 Introduction 1.4 Théorèmes de
Page 34 and 35: 34 Introduction dite "de Kolmogorov
Page 36 and 37: 36 Introduction Pour m = n − 1, u
Page 38 and 39: 38 Introduction et on considère le
Page 40 and 41: 40 Introduction et des exposants de
Page 42 and 43: 42 Introduction Il reste enfin à s
Page 44 and 45: 44 Introduction Si l’on choisit u
Page 46 and 47: 46 Introduction Marco ([LM05]) qui
Page 48 and 49: 48 Introduction dans un petit voisi
Page 50 and 51: 50 Introduction (voir [Pös82]), et
Page 52 and 53: 52 Résultats et questions Question
Page 54 and 55: 54 Résultats et questions 2.2 De l
Page 56 and 57: 56 Résultats et questions Question
Page 59 and 60: 3.1 - Introduction 59 3 Generic exp
Page 61 and 62: 3.1 - Introduction 61 m is the mult
Page 63 and 64:
3.2 - Statement of results 63 Theor
Page 65 and 66:
3.2 - Statement of results 65 Theor
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3.3 - Proof of Theorem 3.6 67 with
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3.3 - Proof of Theorem 3.6 69 with
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3.3 - Proof of Theorem 3.6 71 and t
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3.3 - Proof of Theorem 3.6 73 Propo
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3.3 - Proof of Theorem 3.6 75 3.3.2
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3.3 - Proof of Theorem 3.6 77 Proof
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3.3 - Proof of Theorem 3.6 79 Now,
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3.3 - Proof of Theorem 3.6 81 (iii)
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3.A - Proof of the normal form 83 P
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3.A - Proof of the normal form 85 w
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3.A - Proof of the normal form 87 a
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3.A - Proof of the normal form 89 P
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3.B - SDM functions 91 with g 0 j =
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3.B - SDM functions 93 any value of
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3.B - SDM functions 95 3.B.2 Preval
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3.B - SDM functions 97 3.B.2.3. As
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4.1 - Introduction and main results
Page 101 and 102:
Page 103 and 104:
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4.2 - Proof of Theorem 4.1 and Theo
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4.2 - Proof of Theorem 4.1 and Theo
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4.3 - Further results and comments
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4.A - Generic assumptions 111 Remar
Page 113 and 114:
4.A - Generic assumptions 113 Of co
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5.1 - Introduction 115 5 Polynomial
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5.2 - Main result 117 by Ck (DR) th
Page 119 and 120:
5.3 - Analytical part 119 is given
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5.3 - Analytical part 121 5.3.2 The
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5.3 - Analytical part 123 Indeed, t
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5.3 - Analytical part 125 which sen
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5.4 - Proof of Theorem 5.2 127 and
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Part III From stability to instabil
Page 131 and 132:
6.1 - Introduction 131 6 Improved e
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6.2 - Main results 133 α > 1. Addi
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6.2 - Main results 135 In fact in t
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6.3 - The analytic case 137 Given a
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6.3 - The analytic case 139 Lemma 6
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6.3 - The analytic case 141 contain
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6.4 - The Gevrey case 143 6.4 The G
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6.4 - The Gevrey case 145 with the
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6.4 - The Gevrey case 147 with suit
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Part IV Results of instability Summ
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7.1 - Introduction 151 7 Optimal ti
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7.2 - Main results 153 systems, as
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7.3 - Construction of the perturbat
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7.3 - Construction of the perturbat
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7.4 - Construction of a symbolic dy
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7.5 - Construction of a pseudo-orbi
Page 179 and 180:
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7.6 - Proof of Theorem 7.2 185 Now
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7.6 - Proof of Theorem 7.2 187 If w
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7.6 - Proof of Theorem 7.2 189 and
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7.A - Time-energy coordinates for t
Page 193 and 194:
8.1 - Introduction 193 8 Time of in
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8.2 - Main result 195 Let us recall
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8.3 - Proof of Theorem 8.1 197 8.3.
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8.3 - Proof of Theorem 8.1 199 j
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8.3 - Proof of Theorem 8.1 201 Henc
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8.3 - Proof of Theorem 8.1 203 whil
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8.3 - Proof of Theorem 8.1 205 Proo
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8.A - Gevrey functions 207 Finally
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References [AA89] V.I. Arnold and A
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REFERENCES 211 [Bou09a] , Arnold di
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REFERENCES 213 [Fas90] F. Fassò, L
Page 215 and 216:
REFERENCES 215 [KT09] K. Khanin and
Page 217 and 218:
REFERENCES 217 [MS04] , Wandering d
Page 219:
REFERENCES 219 [Tre04] , Evolution
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PhD thesis - IAS

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