Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension

Abstract

We prove that for every flat surface $\omega$, the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from $\omega$ exhibit a definite amount of deviation from the correct limit in Birkhoff’s and Oseledets’ theorems is strictly less than 1. This theorem extends a result by Chaika and Eskin who proved that such sets have measure 0. We also prove that the Hausdorff dimension of the directions in which Teichmüller geodesics diverge on average in a stratum is bounded above by 1/2, strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly-mixing IETs with permutation $(d,d-1,\dots ,1)$, where $d\ge 5$ is an odd number, is at least 1/2, thus strengthening a result by Avila and Leguil. Combined with a recent result of Chaika and Masur, this shows that the Hausdorff dimension of this set is exactly 1/2.