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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.06865 |
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| _version_ | 1866929751190405120 |
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| author | Godoy, Yamile Salvai, Marcos |
| author_facet | Godoy, Yamile Salvai, Marcos |
| contents | Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior $U$ of $N$. This induces an outer billiard map $B$ on $U$. We prove that $B$ is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Outer billiards in the complex hyperbolic plane Godoy, Yamile Salvai, Marcos Dynamical Systems Differential Geometry 32Q15, 37C83, 53C22, 53C35, 53D22 Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior $U$ of $N$. This induces an outer billiard map $B$ on $U$. We prove that $B$ is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space. |
| title | Outer billiards in the complex hyperbolic plane |
| topic | Dynamical Systems Differential Geometry 32Q15, 37C83, 53C22, 53C35, 53D22 |
| url | https://arxiv.org/abs/2503.06865 |