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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2507.15870 |
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| _version_ | 1866918100994097152 |
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| author | Wei, Yuming |
| author_facet | Wei, Yuming |
| contents | We study the ergodic properties of the translation surface $X_{λ,μ}$ formed by gluing two flat tori along a slit with holonomy $(λ,μ) \in \mathbb{R}^2$. Extending the dichotomy result of Cheung, Hubert, and Masur for the case $μ= 0$, we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set of nonergodic directions is either $0$ or $\frac{1}{2}$. This dichotomy is completely characterized by the Pérez-Marco condition expressed in terms of best approximation denominators. As a corollary, we obtain that the Pérez-Marco condition for best approximation denominators is norm-independent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15870 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dichotomy for the Hausdorff dimension of nonergodic directions on translation surfaces Wei, Yuming Dynamical Systems We study the ergodic properties of the translation surface $X_{λ,μ}$ formed by gluing two flat tori along a slit with holonomy $(λ,μ) \in \mathbb{R}^2$. Extending the dichotomy result of Cheung, Hubert, and Masur for the case $μ= 0$, we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set of nonergodic directions is either $0$ or $\frac{1}{2}$. This dichotomy is completely characterized by the Pérez-Marco condition expressed in terms of best approximation denominators. As a corollary, we obtain that the Pérez-Marco condition for best approximation denominators is norm-independent. |
| title | Dichotomy for the Hausdorff dimension of nonergodic directions on translation surfaces |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2507.15870 |