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

Abstract

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 $\frac{1}{2}$.