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arXiv · 2507.15870

Dichotomy for the Hausdorff dimension of nonergodic directions on translation surfaces

Yuming Wei
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Abstract

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.

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Yuming Wei. 2025-07-16. Dichotomy for the Hausdorff dimension of nonergodic directions on translation surfaces. https://arxiv.org/abs/2507.15870

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