Positive measure of KAM tori for finitely differentiable Hamiltonians

Bounemoura, Abed

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

Soient un entier $n\ge 2$ et des réels $\tau >n-1$ et $\ell >2(\tau +1)$. En utilisant des idées de Moser, Salamon a prouvé que les tores diophantiens individuels persistent pour les systèmes hamiltoniens de classe $C^{\ell }$. Sous l’hypothèse plus forte que le système est une perturbation de classe $C^{\ell +\tau }$ 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 $C^{\ell }$ et que la partie intégrable soit de classe $C^{\ell +2}$. 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 $n\ge 2$ and real numbers $\tau >n-1$ and $\ell >2(\tau +1)$. Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^{\ell }$. Under the stronger assumption that the system is a $C^{\ell +\tau }$ 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 $C^{\ell }$ and the integrable part to be of class $C^{\ell +2}$. 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.

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

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