#### Volume 19, issue 6 (2015)

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

### Jayadev S Athreya and Jon Chaika

Geometry & Topology 19 (2015) 3537–3563
##### Abstract

We show that for almost every (with respect to Masur–Veech measure) translation surface $\omega \in \mathsc{ℋ}\left(2\right)$, 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 $\frac{1}{2}$.

##### Keywords
interval exchange transformation, Hausdorff dimension, Rauzy induction
##### Mathematical Subject Classification 2010
Primary: 37E05, 37E35