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

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 conjectur...

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Bibliographic Details
Published in:arXiv.org 2021-05
Main Authors: Abed Bounemoura, Farré, Gerard
Format: Article
Language:English
Subjects:
Hamiltonian functions
Initial conditions
Invariants
Toruses
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2105.01297