Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.13214 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917972296073216 |
|---|---|
| author | Barbieri, Santiago Clarke, Andrew |
| author_facet | Barbieri, Santiago Clarke, Andrew |
| contents | In this paper we consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map $T$ of the annulus $\mathbb A = \mathbb T \times (0,π)$. We prove the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary $\partial \mathbb A$; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of $\partial \mathbb A$ through which there passes no invariant essential curve; and the only coin billiard for which the phase space $\mathbb A$ is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, we describe the results of some numerical experiments on the elliptical coin billiard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and Nonexistence of Invariant Curves of Coin Billiards Barbieri, Santiago Clarke, Andrew Dynamical Systems 37A05 In this paper we consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map $T$ of the annulus $\mathbb A = \mathbb T \times (0,π)$. We prove the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary $\partial \mathbb A$; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of $\partial \mathbb A$ through which there passes no invariant essential curve; and the only coin billiard for which the phase space $\mathbb A$ is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, we describe the results of some numerical experiments on the elliptical coin billiard. |
| title | Existence and Nonexistence of Invariant Curves of Coin Billiards |
| topic | Dynamical Systems 37A05 |
| url | https://arxiv.org/abs/2411.13214 |