The Hausdorff dimension of non-uniquely ergodic directions in H(2) is almost everywhere 1∕2

Athreya, Jayadev S¹ ; Chaika, Jonathan²

¹ Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350 USA
² Department of Mathematics, University of Utah, 155 S 1400 E Room 233, Salt Lake City, UT 84112-0090, USA

Geometry & topology, Tome 19 (2015) no. 6, pp. 3537-3563.

Résumé

We show that for almost every (with respect to Masur–Veech measure) translation surface $\omega \in \mathcal{\mathscr{H}}\left(2\right)$, the set of angles $\theta \in \left[0,2\pi \right)$ such that $e^{i\theta }\omega$ has non-uniquely ergodic vertical foliation has Hausdorff dimension (and codimension) $\frac{1}{2}$. We show this by proving that the Hausdorff codimension of the set of non-uniquely ergodic interval exchange transformations (IETs) in the Rauzy class of $\left(4321\right)$ is also $\frac{1}{2}$.

Détail

DOI : 10.2140/gt.2015.19.3537

Classification : 37E05, 37E35
Keywords: interval exchange transformation, Hausdorff dimension, Rauzy induction

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