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Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension

Hamid Al-Saqban, Paul Apisa, Alena Erchenko, Osama Khalil, Shahriar Mirzadeh, Caglar Uyanik

Abstract

We prove that for every flat surface $ω$, the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from $ω$ 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 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 $d$ is an odd number, is exactly $1/2$ and strengthen a result by Avila and Leguil.

Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension

Abstract

We prove that for every flat surface , the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from exhibit a definite amount of deviation from the correct limit in Birkhoff's and Oseledets' Theorems is strictly less than . This theorem extends a result by Chaika and Eskin where they proved that such sets have measure . We also prove that the Hausdorff dimension of the directions in which Teichmüller geodesics diverge on average in a stratum is bounded above by , strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly mixing IETs with permutation , where is an odd number, is exactly and strengthen a result by Avila and Leguil.

Paper Structure

This paper contains 39 sections, 52 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction to horocycles
  4. Properties of Hausdorff dimension
  5. Proof of Corollary \ref{['cor: Birkhoff corollary']}
  6. The Contraction Hypothesis and Analysis of Recurrence
  7. Estimates for integrals over horocycle orbits
  8. Coverings and Long Excursions
  9. Proof of Theorem \ref{['thrm: H.dim of non-recurrent directions,fixed compact set']}
  10. Hausdorff Dimension of The Divergent on Average Directions
  11. Uniformity in Birkhoff's Theorem
  12. Some Finiteness and Recurrence Results
  13. Effective Unipotent Invariance
  14. Proof of Theorem \ref{['thrm: quantitative Chaika-Eskin']}
  15. Proof of Theorem \ref{['thrm: quantitative discrete Chaika-Eskin']}
  16. ...and 24 more sections

Key Result

Theorem \oldthetheorem

Suppose $\mathcal{M}\subseteq \mathcal{H}_1(\alpha)$ is an affine invariant submanifold and $\nu_{\mathcal{M}}$ is the affine measure whose support is $\mathcal{M}$. Then, for any bounded continuous function $f$ on $\mathcal{M}$ and any $\varepsilon>0$, there exist affine invariant submanifolds $\ma is at most $\delta$.

Theorems & Definitions (85)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Definition : Strongly Irreducible Cocycle
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem: Analogue of Theorem \ref{['thrm: Birkhoff deviations thrm']}
  • ...and 75 more