Dichotomy for the Hausdorff dimension of non-ergodic directions on translation surfaces
Abstract
Abstract We study the ergodic properties of the translation surface $X_{\unicode{x3bb} ,\mu }$ formed by gluing two flat tori along a slit with holonomy $(\unicode{x3bb} ,\mu ) \in \mathbb {R}^2$ . Extending the dichotomy result of Cheung, Hubert, and Masur for the case $\mu = 0$ , we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set $\operatorname {\mathrm {NE}}(X_{\unicode{x3bb} ,\mu },\omega )$ of non-ergodic directions is either $0$ or $\frac 12$ . 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.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Geometry and complex manifolds
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Ergodic Theory and Dynamical Systems
- Volume
- 46
- Issue
- 5
- Pages
- 1323–1364
- Publication date
- 7 January 2026
- Language
- EN