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

Bounemoura, Abed; Farré, Gerard (2021), Positive measure of effective quasi-periodic motion near a Diophantine torus. https://basepub.dauphine.psl.eu/handle/123456789/23285

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2105.01297.pdf (177.6Kb)
Type
Document de travail / Working paper
Date
2021
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
17
Metadata
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Author(s)
Bounemoura, Abed
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Farré, Gerard
Department of Mathematics [KTH Royal Institute of Technology]
Abstract (EN)
It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus T0, 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 T0 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 T0. 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.
Subjects / Keywords
Hamiltonian systems; quasi-periodic invariant tori; effective stability

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