Positive measure of KAM tori for finitely differentiable Hamiltonians

Abed Bounemoura

doi:10.5802/jep.137

Positive measure of KAM tori for finitely differentiable Hamiltonians
[Mesure positive de tores invariants pour les systèmes hamiltoniens en différentiabilité finie]

Abed Bounemoura ¹

¹ CNRS - PSL Research University, (Université Paris-Dauphine and Observatoire de Paris) Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1113-1132.

Résumé
Abstract

Soient un entier $n \geq 2$ et des réels $τ > n - 1$ et $ℓ > 2 (τ + 1)$ . En utilisant des idées de Moser, Salamon a prouvé que les tores diophantiens individuels persistent pour les systèmes hamiltoniens de classe $C^{ℓ}$ . Sous l’hypothèse plus forte que le système est une perturbation de classe $C^{ℓ + τ}$ d’un système analytique intégrable, Pöschel a prouvé la persistance d’un ensemble de mesure positive de tores diophantiens. Nous améliorons le dernier résultat en montrant qu’il suffit que la perturbation soit de classe $C^{ℓ}$ et que la partie intégrable soit de classe $C^{ℓ + 2}$ . La principale nouveauté consiste à montrer que l’on peut contrôler la régularité Lipschitz par rapport aux fréquences des tores invariants sans contrôler la régularité Lipschitz de la dynamique quasi-périodique sur chaque tore.

Consider an integer $n \geq 2$ and real numbers $τ > n - 1$ and $ℓ > 2 (τ + 1)$ . Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^{ℓ}$ . Under the stronger assumption that the system is a $C^{ℓ + τ}$ perturbation of an analytic integrable system, Pöschel proved the persistence of a set of positive measure of Diophantine tori. We improve the latter result by showing it is sufficient for the perturbation to be of class $C^{ℓ}$ and the integrable part to be of class $C^{ℓ + 2}$ . The main novelty consists in showing that one can control the Lipschitz regularity, with respect to frequency parameters, of the invariant torus, without controlling the Lipschitz regularity of the quasi-periodic dynamics on the invariant torus.

Reçu le : 2019-09-12
Accepté le : 2020-09-16
Publié le : 2020-09-18

Zbl

DOI : 10.5802/jep.137

Classification : 37J40, 37C55
Keywords: Hamiltonian systems, KAM theory
Mot clés : Systèmes hamiltoniens, théorie KAM

Affiliations des auteurs :

Abed Bounemoura ¹

¹ CNRS - PSL Research University, (Université Paris-Dauphine and Observatoire de Paris) Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Positive measure of {KAM} tori for finitely~differentiable {Hamiltonians}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Abed Bounemoura. Positive measure of KAM tori for finitely differentiable Hamiltonians. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 1113-1132. doi : 10.5802/jep.137. https://jep.centre-mersenne.org/articles/10.5802/jep.137/

Bibliographie
Cité par

[Alb07] J. Albrecht - “On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations”, Regul. Chaotic Dyn. 12 (2007) no. 3, p. 281-320 | DOI | MR | Zbl

[Arn63] V. I. Arnold - “Proof of a theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations”, Russian Math. Surveys 18 (1963) no. 5, p. 9-36 | DOI

[BF19] A. Bounemoura & J. Féjoz - “KAM, $α$ -Gevrey regularity and the $α$ -Bruno-Rüssmann condition”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 19 (2019) no. 4, p. 1225-1279 | Zbl

[CW13] C.-Q. Cheng & L. Wang - “Destruction of Lagrangian torus for positive definite Hamiltonian systems”, Geom. Funct. Anal. 23 (2013) no. 3, p. 848-866 | DOI | MR | Zbl

[Her86] M.-R. Herman - Sur les courbes invariantes par les difféomorphismes de l’anneau, Astérisque, vol. 144, Société Mathématique de France, 1986 | Zbl

[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), p. 527-530

[Laz73] V. F. Lazutkin - “Existence of caustics for the billiard problem in a convex domain”, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), p. 186-216 | MR | Zbl

[Mos62] J. Moser - “On invariant curves of area-preserving mappings of an annulus”, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1962), p. 1-20 | MR | Zbl

[Mos70] J. Moser - “On the construction of almost periodic solutions for ordinary differential equations”, in Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, p. 60-67

[Pop04] G. Popov - “KAM theorem for Gevrey Hamiltonians”, Ergodic Theory Dynam. Systems 24 (2004) no. 5, p. 1753-1786 | DOI | MR | Zbl

[Pö80] J. Pöschel - Über invariante tori in differenzierbaren Hamiltonschen systemen, Bonn. Math. Schr., vol. 120, Univ. Bonn, Mathematisches Institut, 1980 | Zbl

[Pö82] J. Pöschel - “Integrability of Hamiltonian systems on Cantor sets”, Comm. Pure Appl. Math. 35 (1982) no. 5, p. 653-696 | DOI | MR | Zbl

[Pö01] J. Pöschel - “A lecture on the classical KAM theory”, in Smooth ergodic theory and its applications (Seattle, WA, 1999) (A. Katok & al., eds.), Proc. Symp. Pure Math., vol. 69, American Mathematical Society, Providence, RI, 2001, p. 707-732 | DOI

[Rü01] H. Rüssmann - “Invariant tori in non-degenerate nearly integrable Hamiltonian systems”, Regul. Chaotic Dyn. 6 (2001) no. 2, p. 119-204 | DOI | MR | Zbl

[Sal04] D. A. Salamon - “The Kolmogorov-Arnold-Moser Theorem”, Math. Phys. Electr. J. 10 (2004), p. 1-37 | MR | Zbl

[Zeh75] E. Zehnder - “Generalized implicit function theorems with applications to some small divisor problems. I”, Comm. Pure Appl. Math. 28 (1975), p. 91-140 | DOI | MR | Zbl

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