Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension
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arXiv
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| Format: | Preprint |
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2017
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| author | Al-Saqban, Hamid Apisa, Paul Erchenko, Alena Khalil, Osama Mirzadeh, Shahriar Uyanik, Caglar |
| author_facet | Al-Saqban, Hamid Apisa, Paul Erchenko, Alena Khalil, Osama Mirzadeh, Shahriar Uyanik, Caglar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1711_10542 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension Al-Saqban, Hamid Apisa, Paul Erchenko, Alena Khalil, Osama Mirzadeh, Shahriar Uyanik, Caglar Dynamical Systems Geometric Topology 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. |
| title | Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension |
| topic | Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/1711.10542 |