Bialy-Mironov type rigidity for centrally symmetric symplectic billiards
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916142151368704 |
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| author | Baracco, Luca Bernardi, Olga Nardi, Alessandra |
| author_facet | Baracco, Luca Bernardi, Olga Nardi, Alessandra |
| contents | The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic billiard map has a (simple) continuous invariant curve $δ\subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial D$) and consisting only of $4$-periodic orbits. If one of the parts between $δ$ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then $\partial D$ is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve $δ$, and the result is obtained by tracing back the problem directly to the totally integrable case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bialy-Mironov type rigidity for centrally symmetric symplectic billiards Baracco, Luca Bernardi, Olga Nardi, Alessandra Dynamical Systems 37E40 37J35 The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic billiard map has a (simple) continuous invariant curve $δ\subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial D$) and consisting only of $4$-periodic orbits. If one of the parts between $δ$ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then $\partial D$ is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve $δ$, and the result is obtained by tracing back the problem directly to the totally integrable case. |
| title | Bialy-Mironov type rigidity for centrally symmetric symplectic billiards |
| topic | Dynamical Systems 37E40 37J35 |
| url | https://arxiv.org/abs/2402.19154 |