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Dichotomy for the Hausdorff dimension of non-ergodic directions on translation surfaces
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 46 / Issue 5 / May 2026
- Published online by Cambridge University Press:
- 07 January 2026, pp. 1323-1364
- Print publication:
- May 2026
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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.
