Chaotic properties for billiards in circular polygons
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916435657228288 |
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| author | Clarke, Andrew Ramírez-Ros, Rafael |
| author_facet | Clarke, Andrew Ramírez-Ros, Rafael |
| contents | We study billiards in domains enclosed by circular polygons. These are closed $C^1$ 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 $N \in \mathbb{N}$, 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 $q \to \infty$, and present an unusual property of the length spectrum. Our proofs are entirely analytical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09892 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Chaotic properties for billiards in circular polygons Clarke, Andrew Ramírez-Ros, Rafael Dynamical Systems Chaotic Dynamics 37C83 (Primary), 37B10 (Secondary) We study billiards in domains enclosed by circular polygons. These are closed $C^1$ 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 $N \in \mathbb{N}$, 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 $q \to \infty$, and present an unusual property of the length spectrum. Our proofs are entirely analytical. |
| title | Chaotic properties for billiards in circular polygons |
| topic | Dynamical Systems Chaotic Dynamics 37C83 (Primary), 37B10 (Secondary) |
| url | https://arxiv.org/abs/2309.09892 |