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...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/421747 |
| Acesso em linha: | https://hdl.handle.net/2117/421747 https://dx.doi.org/10.1007/s00220-024-05113-4 |
| Access Level: | acceso abierto |
| Palavra-chave: | 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 |
| Resumo: | 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. |
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