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,\ldots,1) , where d \geq 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.

Divergent on average directions of Teichmüller geodesic flow

“…For any quadratic differential the set of directions that diverge on average in Q g;n is contained in the set of directions that diverge on average in the stratum. In al-Saqban, Apisa, Erchenko, Khalil, Mirzadeh and Uyanik [1], the authors adapted the techniques of Kadyrov, Kleinbock, Lindenstrauss and Margulis [12] to show that the latter set has Hausdorff dimension at most 1 2 (this result improves on results of Masur [17,18]). Therefore, the novelty of the current work is establishing the lower bound of Hausdorff dimension 1 2 for the set of directions that diverge on average in Q g;n .…”

“…Theorem 1. For a quadratic or Abelian differential q the set of directions  2 OE0; 2 such that the Teichmüller geodesic ¹g t r  qº t 0 determined by r  q diverges on average (either in its stratum or in Q g;n ) has Hausdorff dimension exactly equal to 1 2 . Given a noncompact Hausdorff topological space , let C 0 .…”

“…Remark 3. The methods of [1] in fact show that the Hausdorff dimension of the set of directions that diverge on average in any open SL.2; R/ invariant subset of a stratum is at most 1 2 . Therefore, Theorem 1 remains true when divergence on average is considered on any open SL.2; R/ invariant subset of a stratum of quadratic differentials.…”

Weakly Mixing Polygonal Billiards

“…Lemma 3.9. [AAEKMU,Lemma 3.5] Let f M be as in Theorem 3.8. Then there exists a constant b ′ > 0 so that for all 0 < a < 1 there exists t0 = t0 (a) such that for all t > t0 and for all ω ∈ H \ M we have…”

“…Proposition 3.10. [AAEKMU,Proposition 3.7] Let f M be as in Theorem 3.8 and let b ′ > 0 and t0 = t0 (a) be as in Lemma 3.9. There exist C 1 > 1 (independent of ω and a) such that for all a ∈ (0, 1), all ρ > C 1 b ′ /a, all t ≥ t0 such that e t ∈ N and all N ∈ N,…”