PhD thesis - IAS

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214 REFERENCES [HPS77] M.W. Hirsch, C.C. Pugh, and M. Shub, Invariant Manifolds, Springer, Berlin, 1977. [HSY92] B.R. Hunt, T. Sauer, and J.A. Yorke, Prevalence: a translation-invariant “almost every" on infinite-dimensional spaces, Bull. of the Amer. Math. Soc. 27 (1992), 217–238. [HZ94] H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts Verlag, Basel, 1994. [Ily86] I.S. Ilyashenko, A steepness test for analytic functions, Russian Math. Surveys 41 (1986), 229–230. [JV97] À. Jorba and J. Villanueva, On the normal behaviour of partially elliptic lower dimensional tori of Hamiltonian systems, Nonlinearity 10 (1997), 783– 822. [KH08] V. Kaloshin and B. Hunt, Prevalence, to appear in Handbook of Dynamical Systems, Volume 3, edited by Henk Broer, Boris Hasselblatt and Floris Takens, 2008. [KL08a] V. Kaloshin and M. Levi, An example of Arnold diffusion for near-integrable Hamiltonians, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 3, 409–427. [KL08b] , Geometry of Arnold diffusion, SIAM Rev. 50 (2008), no. 4, 702–720. [KLDM06] K. Khanin, J. Lopes Dias, and J. Marklof, Renormalization of multidimensional Hamiltonian flows, Nonlinearity 19 (2006), no. 12, 2727–2753. [KLDM07] , Multidimensional continued fractions, dynamical renormalization and KAM theory, Comm. Math. Phys. 270 (2007), no. 1, 197–231. [KLS10] V. Kaloshin, M. Levi, and M. Saprykina, An example of nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension, Preprint (2010). [KMV04] V. Kaloshin, J. N. Mather, and E. Valdinoci, Instability of resonant totally elliptic points of symplectic maps in dimension 4, Loday-Richaud, Michèle (ed.), Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II. Volume en l’honneur de Jean-Pierre Ramis. Paris: Société Mathématique de France. Astérisque 297, 79-116 , 2004. [Kol54] A.N. Kolmogorov, On the preservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk. SSSR 98 (1954), 527–530. [Koz96] V.V. Kozlov, Symmetries, Topology, and Resonance in Hamiltonian Mechanics, Springer-Verlag, Berlin, Heidelberg, 1996. [KP94] S. Kuksin and J. Pöschel, On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications, Seminar on dynamical systems (1994), 96–116, Birkhäuser, Basel.
REFERENCES 215 [KT09] K. Khanin and A. Teplinsky, Herman’s theory revisited, Invent. Math. 178 (2009), no. 2, 333–344. [KZZ09] V. Kaloshin, K. Zhang, and Y. Zheng, Almost dense orbit on energy surface, Proceedings of the XVI-th ICMP, Prague, 314-322, 2009. [Lan02] S. Lang, Algebra, Graduate Texts in Mathematics 211, Springer Verlag, New-York, 2002. [LM88] P. Lochak and C. Meunier, Multiphase averaging for classical systems. With applications to adiabatic theorems. Transl. from the French by H. S. Dumas, Applied Mathematical Sciences, 72, New York etc, Springer-Verlag. xi, 360 pp., 1988. [LM01] M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, Katok, Anatole (ed.) et al., Smooth ergodic theory and its applications (Seattle, WA, 1999). Providence, RI: Amer. Math. Soc. (AMS). Proc. Symp. Pure Math. 69, 733-746, 2001. [LM05] P Lochak and J.P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Central European Journal of Mathematics 3 (2005), no. 3, 342–397. [LMS03] P. Lochak, J.-P. Marco, and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Am. Math. Soc. 775 (2003), 145 pp. [LN92] P. Lochak and A.I. Neishtadt, Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian, Chaos 2 (1992), no. 4, 495–499. [LNN94] P. Lochak, A.I. Neistadt, and L. Niederman, Stability of nearly integrable convex Hamiltonian systems over exponentially long times., Kuksin, S. (ed.) et al., Seminar on dynamical systems. Basel: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 12, 15-34, 1994. [Loc92] P. Lochak, Canonical perturbation theory via simultaneous approximation., Russ. Math. Surv. 47 (1992), no. 6, 57–133. [Loc93] , Hamiltonian perturbation theory: periodic orbits, resonances and intermittency., Nonlinearity 6 (1993), no. 6, 885–904. [Loc95] Pierre Lochak, Stability of Hamiltonian systems over exponentially long times: The near-linear case, Dumas, H. S. (ed.) et al., Hamiltonian dynamical systems: history, theory, and applications. Proceedings of the international conference held at the University of Cincinnati, OH (USA), March 1992. New York, NY: Springer-Verlag. IMA Vol. Math. Appl. 63, 221-229, 1995. [Loc99] , Arnold diffusion ; a compendium of remarks and questions., Simó, Carles (ed.), Hamiltonian systems with three or more degrees of freedom. Proceedings of the NATO Advanced Study Institute, 1995. Dordrecht: Kluwer Academic Publishers., 1999.
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N o d’ordre: 9910 THÈSE Présent
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Résumé Dans cette thèse, on s’
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Béton style ... Quelque soit l’
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Remerciements À mes deux excellent
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5.2 Main result . . . . . . . . . .
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1.1 - Systèmes hamiltoniens 17 1 I
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1.1 - Systèmes hamiltoniens 19 où
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1.1 - Systèmes hamiltoniens 21 man
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1.2 - Systèmes hamiltoniens intég
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1.2 - Systèmes hamiltoniens intég
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1.3 - Théorie classique des pertur
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1.5 - Exemples d’instabilité 41
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1.6 - Au voisinage d’un tore inva
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1.6 - Au voisinage d’un tore inva
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2.1 - Résultats de stabilité 51 2
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2.1 - Résultats de stabilité 53 R
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2.3 - Résultats d’instabilité 5
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Part II Results of stability Summar
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60 Generic exponential stability wi
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116 Polynomial stability for C k qu
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132 Improved exponential stability
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152 Optimal time of instability for
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Page 210 and 211: 210 REFERENCES [Ber09] , Large norm
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