The Hausdorff Dimension of Divergent on Average Sets of Directions for Teichmuller Geodesic Flow.

Group Actions and Dynamics
Event time: 
Monday, October 1, 2018 - 4:15pm
LOM 206
Paul Apisa
Speaker affiliation: 
Event description: 

Abstract: Given a translation surface there is a circle of directions in which one might apply Teichmuller geodesic flow. We will describe work showing that for every (not just almost every!) translation surface the set of directions in which Teichmuller geodesic flow diverges on average - i.e. spends asymptotically zero percent of its time in any compact set - is 1/2. 

In the first part of the talk, we will recall work of Masur, which connects divergence of Teichmuller geodesic flow with non-unique ergodicity of straight-line flow on the translation surface. We will then review the continued fraction case and mention related work of Kadyrov, Kleinbock, Lindenstrauss, and Margulis in the homogenous setting. 

In the second part of the talk, we will describe the lower bound (joint with H. Masur) and how it depends on (1) a quantitative recurrence result for Teichmuller geodesic flow and (2) the quadratic growth of cylinders on translation surfaces.

In the third and final part of the talk, we will describe the upper bound (joint with H. al-Saqban, A. Erchenko, O. Khalil, S. Mirzadeh, and C. Uyanik), which adapts the argument of Kadyrov, Kleinbock, Lindenstrauss, and Margulis to the Teichmuller geodesic flow setting using Margulis functions. Time permitting we will also describe work which shows that for any compactly-supported continuous function the set of directions for which the time-averages along the flow deviate from the space-average by a fixed definite amount have Hausdorff dimension strictly less than one.