Positive measure of KAM tori for finitely differentiable Hamiltonians
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 1113-1132.

Consider an integer n2 and real numbers τ>n-1 and >2(τ+1). Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class C . Under the stronger assumption that the system is a C +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 and the integrable part to be of class C +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.

Soient un entier n2 et des réels τ>n-1 et >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 . Sous l’hypothèse plus forte que le système est une perturbation de classe C +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 et que la partie intégrable soit de classe C +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.

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DOI: 10.5802/jep.137
Classification: 37J40, 37C55
Keywords: Hamiltonian systems, KAM theory
Mot clés : Systèmes hamiltoniens, théorie KAM

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Abed Bounemoura. Positive measure of KAM tori for finitely differentiable Hamiltonians. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 1113-1132. doi : 10.5802/jep.137. https://jep.centre-mersenne.org/articles/10.5802/jep.137/

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