Expansiveness and hyperbolicity in convex billiards

Detalhes bibliográficos
Autor(a) principal: Bessa, Mário
Data de Publicação: 2021
Outros Autores: Dias, João Lopes, Torres, M. J.
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: https://hdl.handle.net/1822/75225
Resumo: We say that a convex planar billiard table B is C^2-stably expansive on a fixed open subset U of the phase space if its billiard map f_B is expansive on the maximal invariant set Λ_(B,U) = ∩_(n∈Z ) f_B^n(U), and this property holds under C^2 perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of f_B in Λ_(B,U) is uniformly hyperbolic. In addition we show that this property also holds for a generic choice among billiards which are expansive.
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spelling Expansiveness and hyperbolicity in convex billiardsConvex planar billiardsHyperbolic setsExpansivenessCiências Naturais::MatemáticasScience & TechnologyWe say that a convex planar billiard table B is C^2-stably expansive on a fixed open subset U of the phase space if its billiard map f_B is expansive on the maximal invariant set Λ_(B,U) = ∩_(n∈Z ) f_B^n(U), and this property holds under C^2 perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of f_B in Λ_(B,U) is uniformly hyperbolic. In addition we show that this property also holds for a generic choice among billiards which are expansive.The authors were partially funded by the project "New Trends in Lyapunov Exponents" PTDC/MAT-PUR/29126/2017 financed by Fundacao para a Ciencia e a Tecnologia, Portugal.MB would also like to thank CMUP for providing the necessary conditions under which this work was developed. JLD was also partially funded by the Project CEMAPRE-UID/MULTI/00491/2019 financed by Fundacao para a Ciencia e a Tecnologia. MJT was also partially financed by national funds through Fundacao para a Ciencia e a Tecnologia within the projects UIDB/00013/2020 and UIDP/00013/2020.SpringerUniversidade do MinhoBessa, MárioDias, João LopesTorres, M. J.20212021-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/1822/75225eng1560-35471468-484510.1134/S1560354721060125https://link.springer.com/article/10.1134/S1560354721060125info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-21T12:11:54Zoai:repositorium.sdum.uminho.pt:1822/75225Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T19:03:44.682942Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Expansiveness and hyperbolicity in convex billiards
title Expansiveness and hyperbolicity in convex billiards
spellingShingle Expansiveness and hyperbolicity in convex billiards
Bessa, Mário
Convex planar billiards
Hyperbolic sets
Expansiveness
Ciências Naturais::Matemáticas
Science & Technology
title_short Expansiveness and hyperbolicity in convex billiards
title_full Expansiveness and hyperbolicity in convex billiards
title_fullStr Expansiveness and hyperbolicity in convex billiards
title_full_unstemmed Expansiveness and hyperbolicity in convex billiards
title_sort Expansiveness and hyperbolicity in convex billiards
author Bessa, Mário
author_facet Bessa, Mário
Dias, João Lopes
Torres, M. J.
author_role author
author2 Dias, João Lopes
Torres, M. J.
author2_role author
author
dc.contributor.none.fl_str_mv Universidade do Minho
dc.contributor.author.fl_str_mv Bessa, Mário
Dias, João Lopes
Torres, M. J.
dc.subject.por.fl_str_mv Convex planar billiards
Hyperbolic sets
Expansiveness
Ciências Naturais::Matemáticas
Science & Technology
topic Convex planar billiards
Hyperbolic sets
Expansiveness
Ciências Naturais::Matemáticas
Science & Technology
description We say that a convex planar billiard table B is C^2-stably expansive on a fixed open subset U of the phase space if its billiard map f_B is expansive on the maximal invariant set Λ_(B,U) = ∩_(n∈Z ) f_B^n(U), and this property holds under C^2 perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of f_B in Λ_(B,U) is uniformly hyperbolic. In addition we show that this property also holds for a generic choice among billiards which are expansive.
publishDate 2021
dc.date.none.fl_str_mv 2021
2021-01-01T00:00:00Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv https://hdl.handle.net/1822/75225
url https://hdl.handle.net/1822/75225
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 1560-3547
1468-4845
10.1134/S1560354721060125
https://link.springer.com/article/10.1134/S1560354721060125
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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