Invariant sets for the wind-tree model
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917006991687680 |
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| author | Tumarkin, Yuriy |
| author_facet | Tumarkin, Yuriy |
| contents | We consider the wind-tree model, a $\mathbb{Z}^2$ - periodic billiard. In the case when the underlying compact translation surface lies on a periodic orbit of the Teichmüller geodesic flow, and at least one of the two homology classes defining the $\mathbb{Z}^2$ - cover is unstable for the Kontsevich-Zorich cocycle, we prove that every orbit closure of the billiard has Hausdorff dimension strictly smaller than 2. The proof relies on a construction of explicit invariant functions, which along the way gives a new proof of non-ergodicity and non-transitivity of the wind-tree model for all parameters and almost all directions, as first shown by Frcaczek and Ulcigrai (2014). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant sets for the wind-tree model Tumarkin, Yuriy Dynamical Systems We consider the wind-tree model, a $\mathbb{Z}^2$ - periodic billiard. In the case when the underlying compact translation surface lies on a periodic orbit of the Teichmüller geodesic flow, and at least one of the two homology classes defining the $\mathbb{Z}^2$ - cover is unstable for the Kontsevich-Zorich cocycle, we prove that every orbit closure of the billiard has Hausdorff dimension strictly smaller than 2. The proof relies on a construction of explicit invariant functions, which along the way gives a new proof of non-ergodicity and non-transitivity of the wind-tree model for all parameters and almost all directions, as first shown by Frcaczek and Ulcigrai (2014). |
| title | Invariant sets for the wind-tree model |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2510.10760 |