Chaotic properties of billiards in circular polygons

We study billiards in domains enclosed by circular polygons. These are closed strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories close enough to the boundary of the domain, in which the return bi...

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Detalles Bibliográficos
Autores: Clarke, Andrew Michael|||0000-0002-9141-4158, Ramírez Ros, Rafael|||0000-0002-2127-2940
Tipo de recurso: artículo
Fecha de publicación:2024
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/421747
Acceso en línea:https://hdl.handle.net/2117/421747
https://dx.doi.org/10.1007/s00220-024-05113-4
Access Level:acceso abierto
Palabra clave:Differentiable dynamical systems
Billiards
Circular polygons
Chaos
Symbolic dynamics
Periodic trajectories
Length spectrum
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
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spelling Chaotic properties of billiards in circular polygonsClarke, Andrew Michael|||0000-0002-9141-4158Ramírez Ros, Rafael|||0000-0002-2127-2940Differentiable dynamical systemsBilliardsCircular polygonsChaosSymbolic dynamicsPeriodic trajectoriesLength spectrumSistemes dinàmics diferenciablesClassificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theoryÀrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmicsWe study billiards in domains enclosed by circular polygons. These are closed strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories close enough to the boundary of the domain, in which the return billiard dynamics is semiconjugate to a transitive subshift on infinitely many symbols that contains the full N-shift as a topological factor for any , so it has infinite topological entropy. We prove the existence of uncountably many asymptotic generic sliding trajectories approaching the boundary with optimal uniform linear speed, give an explicit exponentially big (in q) lower bound on the number of q-periodic trajectories as , and present an unusual property of the length spectrum. Our proofs are entirely analytical.Peer Reviewed20242024-10-1220252025-01-13journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/421747https://dx.doi.org/10.1007/s00220-024-05113-4reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengEuropean Commission http://doi.org/10.13039/100010661 Horizon 2020 Framework Programme 757802 Instabilities and homoclinic phenomena in Hamiltonian systemsopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4217472026-05-27T15:37:01Z
dc.title.none.fl_str_mv Chaotic properties of billiards in circular polygons
title Chaotic properties of billiards in circular polygons
spellingShingle Chaotic properties of billiards in circular polygons
Clarke, Andrew Michael|||0000-0002-9141-4158
Differentiable dynamical systems
Billiards
Circular polygons
Chaos
Symbolic dynamics
Periodic trajectories
Length spectrum
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
title_short Chaotic properties of billiards in circular polygons
title_full Chaotic properties of billiards in circular polygons
title_fullStr Chaotic properties of billiards in circular polygons
title_full_unstemmed Chaotic properties of billiards in circular polygons
title_sort Chaotic properties of billiards in circular polygons
dc.creator.none.fl_str_mv Clarke, Andrew Michael|||0000-0002-9141-4158
Ramírez Ros, Rafael|||0000-0002-2127-2940
author Clarke, Andrew Michael|||0000-0002-9141-4158
author_facet Clarke, Andrew Michael|||0000-0002-9141-4158
Ramírez Ros, Rafael|||0000-0002-2127-2940
author_role author
author2 Ramírez Ros, Rafael|||0000-0002-2127-2940
author2_role author
dc.subject.none.fl_str_mv Differentiable dynamical systems
Billiards
Circular polygons
Chaos
Symbolic dynamics
Periodic trajectories
Length spectrum
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
topic Differentiable dynamical systems
Billiards
Circular polygons
Chaos
Symbolic dynamics
Periodic trajectories
Length spectrum
Sistemes dinàmics diferenciables
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
description We study billiards in domains enclosed by circular polygons. These are closed strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories close enough to the boundary of the domain, in which the return billiard dynamics is semiconjugate to a transitive subshift on infinitely many symbols that contains the full N-shift as a topological factor for any , so it has infinite topological entropy. We prove the existence of uncountably many asymptotic generic sliding trajectories approaching the boundary with optimal uniform linear speed, give an explicit exponentially big (in q) lower bound on the number of q-periodic trajectories as , and present an unusual property of the length spectrum. Our proofs are entirely analytical.
publishDate 2024
dc.date.none.fl_str_mv 2024
2024-10-12
2025
2025-01-13
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/421747
https://dx.doi.org/10.1007/s00220-024-05113-4
url https://hdl.handle.net/2117/421747
https://dx.doi.org/10.1007/s00220-024-05113-4
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv European Commission http://doi.org/10.13039/100010661 Horizon 2020 Framework Programme 757802 Instabilities and homoclinic phenomena in Hamiltonian systems
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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