Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension

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,…”