Friday, April 26, 2019
04/26/2019 - 1:15pm
This talk is intended as a prologue to the subsequent GATSBY lecture, which will be delivered by Jon Chaika.
I will begin by describing how playing billiards in a polygon with all angles rational multiples of pi, naturally leads to two distinct, but related, dynamical systems - the dynamics of straight line flow on a flat surface and the dynamics of Teichmuller geodesic flow on the moduli space of holomorphic one-forms on Riemann surfaces.
Special attention will be given to the connection between the non-unique-ergodicity of straight line flow and the divergence of Teichmuller geodesics. The talk will conclude with an example of a minimal non-uniquely ergodic foliation of a flat surface.
No background will be assumed for the talk.
04/26/2019 - 2:30pm to 3:30pm
Ratner, Margulis, Dani and many others, showed that the horocycle flow on homogeneous spaces has strong measure theoretic and topological rigidity properties. Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, showed that the action of SL(2,R) and the upper triangular subgroup of SL(2,R) on strata of translation surfaces have similar rigidity properties. We will describe how some of these results fail for the horocycle flow on strata of translation surfaces. In particular,
1) There exist horocycle orbit closures with fractional Hausdorff dimension.
2) There exist points which do not equidistribute under the horocycle flow with respect to any measure.
3) There exist points which equidistribute distribute under the horocycle flow to a measure, but they are not in the topological support of that measure.
No familiarity with these objects will be assumed and the talk will begin with motivating the subject of dynamics and ergodic theory. This is joint work with John Smillie and Barak Weiss.