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Article Dans Une Revue Annales Henri Poincaré Année : 2023

Positive measure of effective quasi-periodic motion near a Diophantine torus

Résumé

It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus $\mathcal{T}_0$, invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian Diophantine quasi-periodic invariant tori. While the conjecture is still open, we will prove the following weaker statement: there exists an open set of positive measure (in fact, the relative measure of the complement is exponentially small) around $\mathcal{T}_0$ such that the motion of all initial conditions in this set is "effectively" quasi-periodic in the sense that they are close to being quasi-periodic for an interval of time which is doubly exponentially long with respect to the inverse of the distance to $\mathcal{T}_0$. This open set can be thought as a neighborhood of a hypothetical invariant set of Lagrangian Diophantine quasi-periodic tori, which may or may not exist.

Dates et versions

hal-03445983 , version 1 (24-11-2021)

Identifiants

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Abed Bounemoura, Gerard Farré. Positive measure of effective quasi-periodic motion near a Diophantine torus. Annales Henri Poincaré, 2023, 24 (9), pp.3289-3304. ⟨10.1007/s00023-023-01302-4⟩. ⟨hal-03445983⟩
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