On Cross Sections to the Horocycle and Geodesic Flows on Quotients by Hecke Triangle Groups

Abstract

The study of continuous dynamical systems via surfaces of section is one of the standard techniques in nonlinear mathematics. This is done by considering the intersections of trajectories in a phase space with a subspace of codimension one. The sought for goal is simplifying the study of the original dynamical system. In this manuscript thesis, we consider cross sections to the horocycle and geodesic flows on quotients of SL(2,R) by the Hecke triangle groups, and applications to Farey statistics and symbolic dynamics.