On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians

It is known that a Diophantine quasi-periodic torus with frequency ω∈Ωτd of a Cl Hamiltonian is effectively stable for a time T(r) that is polynomial on the inverse of the distance to the torus, that we denote by r, with exponent 1 + (l- 2) / (τ+ 1) . It is also known that a Diophantine quasi-period...

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Author: Farré, G.
Format: article
Status:Published version
Publication Date:2023
Country:España
Institution:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repository:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/537445
Online Access:http://hdl.handle.net/2072/537445
Access Level:Open access
Keyword:Effective stability
Hamiltonian systems
Quasi-periodic tori
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spelling On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey HamiltoniansFarré, G.Effective stabilityHamiltonian systemsQuasi-periodic toriIt is known that a Diophantine quasi-periodic torus with frequency ω∈Ωτd of a Cl Hamiltonian is effectively stable for a time T(r) that is polynomial on the inverse of the distance to the torus, that we denote by r, with exponent 1 + (l- 2) / (τ+ 1) . It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian H∈ Gα,L is effectively stable for an exponentially long time on the inverse of the distance to the torus with exponent 1 / (α(1 + τ)) . In this note, we see that following the methods in [11] one can show the almost optimality of these exponents. We also show that, for a dense subset of non-resonant vectors, for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians, the naive lower bound T(r) ≥ Cr- 1 is optimal in terms of the exponent. © 2023, The Author(s).The author is grateful to H. Eliasson, T. Seara, and F. Trujillo for valuable discussions and to the referee for useful comments. This work has been supported by the Juan de la Cierva-Formación Program (FJC2021-044720-I) and the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence (CEX2020-001084-M). It was completed while the author was in residence at Institut Mittag-Leffler in Djursholm, Sweden, and has therefore also been supported by the Swedish Research Council under grant no. 2021-06594.Birkhauser2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion13 p.application/pdfhttp://hdl.handle.net/2072/537445RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésArchiv der MathematikL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5374452026-05-29T05:05:01Z
dc.title.none.fl_str_mv On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
title On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
spellingShingle On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
Farré, G.
Effective stability
Hamiltonian systems
Quasi-periodic tori
title_short On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
title_full On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
title_fullStr On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
title_full_unstemmed On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
title_sort On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians
dc.creator.none.fl_str_mv Farré, G.
author Farré, G.
author_facet Farré, G.
author_role author
dc.subject.none.fl_str_mv Effective stability
Hamiltonian systems
Quasi-periodic tori
topic Effective stability
Hamiltonian systems
Quasi-periodic tori
description It is known that a Diophantine quasi-periodic torus with frequency ω∈Ωτd of a Cl Hamiltonian is effectively stable for a time T(r) that is polynomial on the inverse of the distance to the torus, that we denote by r, with exponent 1 + (l- 2) / (τ+ 1) . It is also known that a Diophantine quasi-periodic torus of a Gevrey Hamiltonian H∈ Gα,L is effectively stable for an exponentially long time on the inverse of the distance to the torus with exponent 1 / (α(1 + τ)) . In this note, we see that following the methods in [11] one can show the almost optimality of these exponents. We also show that, for a dense subset of non-resonant vectors, for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians, the naive lower bound T(r) ≥ Cr- 1 is optimal in terms of the exponent. © 2023, The Author(s).
publishDate 2023
dc.date.none.fl_str_mv 2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/2072/537445
url http://hdl.handle.net/2072/537445
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Archiv der Mathematik
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 13 p.
application/pdf
dc.publisher.none.fl_str_mv Birkhauser
publisher.none.fl_str_mv Birkhauser
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,812455

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