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abstract: 'We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there is a set of almost full positive Lebesgue measure of KAM tori which are doubly exponentially stable. Our results hold true for real-analytic but more generally for Gevrey smooth systems.'
author:
- 'Abed Bounemoura[^1] and Bassam Fayad[^2] and Laurent Niederman[^3]'
bibliography:
- 'DoubleExpTore10.bib'
---

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Introduction and results
========================

Introduction
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The goal of this paper is to prove that invariant Lagrangian Diophantine tori in Hamiltonian systems are generically doubly exponentially stable. We will consider two settings.

In a first setting, we consider a Lagrangian Diophantine torus invariant by a Hamiltonian flow on an arbitrary symplectic manifold. It is well-known that by a symplectic change of coordinates, one can consider a Hamiltonian $$\label{H1}\tag{H}
H(\theta,I)=\omega\cdot I + O\left(||I||^2\right), \quad (\theta,I)\in \T^n \times B,$$ 