Billiards and Hofer's Geometry
Fuente:
arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917070119108608 |
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| author | Berezovik, Mark Kliakhandler, Konstantin Ostrover, Yaron Polterovich, Leonid |
| author_facet | Berezovik, Mark Kliakhandler, Konstantin Ostrover, Yaron Polterovich, Leonid |
| contents | We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. We use this result to show that the billiard ball map of a convex polygon lies in the completion, with respect to Hofer's metric, of the group of smooth area-preserving maps of the annulus. Finally, we discuss related connections to dynamics and pose several open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Billiards and Hofer's Geometry Berezovik, Mark Kliakhandler, Konstantin Ostrover, Yaron Polterovich, Leonid Dynamical Systems Symplectic Geometry 37C83, 53D22 We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. We use this result to show that the billiard ball map of a convex polygon lies in the completion, with respect to Hofer's metric, of the group of smooth area-preserving maps of the annulus. Finally, we discuss related connections to dynamics and pose several open problems. |
| title | Billiards and Hofer's Geometry |
| topic | Dynamical Systems Symplectic Geometry 37C83, 53D22 |
| url | https://arxiv.org/abs/2507.04767 |