Exceptional directions for the Teichm\"{u}ller geodesic flow and Hausdorff dimension

TL;DR

This paper investigates the Hausdorff dimension of sets of directions for Teichmüller geodesics with deviations from classical theorems, showing these sets have dimensions less than 1 and providing bounds and exact codimensions for specific cases.

Contribution

It extends previous measure-zero results to Hausdorff dimension bounds for deviation sets and establishes exact codimension for non-weakly mixing IETs with specific permutations.

Findings

Hausdorff dimension of deviation directions is less than 1

Bound on Hausdorff dimension of diverging directions is 1/2

Codimension of non-weakly mixing IETs with permutation (d, d-1, ..., 1) is exactly 1/2

Abstract

We prove that for every flat surface $\omega$, the Hausdorff dimension of the set of directions in which Teichm\"{u}ller 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 where they proved that such sets have measure $0$. We also prove that the Hausdorff dimension of the directions in which Teichm\"{u}ller 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 $d$ is an odd number, is exactly $1/2$ and strengthen a result by Avila and Leguil.