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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.15870 |
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Table of 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.