Linear Flows on Translation Prisms
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909576551464960 |
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| author | Athreya, Jayadev S. Bédaride, Nicolas Hooper, W. Patrick Hubert, Pascal |
| author_facet | Athreya, Jayadev S. Bédaride, Nicolas Hooper, W. Patrick Hubert, Pascal |
| contents | Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular $n$-gons for $n=7, 9, 14, 16, 18, 20, 24, 30$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear Flows on Translation Prisms Athreya, Jayadev S. Bédaride, Nicolas Hooper, W. Patrick Hubert, Pascal Dynamical Systems 37C83 (Primary) 37E35, 37A25 (Secondary) Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular $n$-gons for $n=7, 9, 14, 16, 18, 20, 24, 30$. |
| title | Linear Flows on Translation Prisms |
| topic | Dynamical Systems 37C83 (Primary) 37E35, 37A25 (Secondary) |
| url | https://arxiv.org/abs/2504.08955 |