Title: Exceptional Directions for the Teichmuller Geodesic Flow
Speaker: Osama Khalil (Ohio State University)
Abstract: A translation surface is a pair (S,w) of a compact Riemann surface S equipped with a holomorphic 1-form w. These objects fit together in a moduli space which admits a natural action of SL(2,R). The work of Masur, Veech and many others has demonstrated that the dynamical behavior of the orbit of a translation surface (S,w) under the diagonal group in SL(2,R) is intimately tied to the ergodic properties of the translation flow induced by the vertical foliation of w on S. However, the hyperbolic nature of this diagonal flow often allows only for an understanding of the behavior of generic points. On the other hand, the remarkable results of Eskin-Mirzakhani-Mohammadi made it possible to understand the dynamical behavior of the full SL(2,R) orbit of every (not just almost every!) such (S,w). In this talk, we show how one can leverage these powerful results to show that the subset of the circle of directions around any fixed (S,w) for which the time-averages along the diagonal flow deviate from the space-average by a definite amount have Hausdorff dimension strictly less than one. We also give a sharp upper bound on the dimension of the directions in which orbits diverge on average. Applications to translation flows and interval exchanges will be discussed. This is joint work with Al-Saqban, Apisa, Erchenko, Uyanik, and Mirzadeh.
Seminar URL: https://research.math.osu.edu/ggt/
