In this talk I will examine how techniques in Teichmüller space yield results about certain dynamical systems. I will also apply probabilistic methods to study the mapping class group of a surface.
I will be interested in folations on surfaces defined by flat structures with singularities. Such flat structures naturally arise in the study of rational billiards, in complex analysis as trajectories of quadratic differentials on Riemann surfaces, and in topology as measured foliations. I will describe how to use Teichmüller maps to study dynamical properties of these foliations. I will focus on two aspects of these dynamical systems. The first is finding periodic orbits. The second is describing ergodic properties.
In the second part of the talk I will discuss random walks on the mapping class group of a surface. I will relate the boundary of the random walk to the sphere of measured foliatons found by Thurston and indicate the role of Teichmüller space in this theory.