Symbolic dynamics for the geodesic flow on Hecke surfaces

TL;DR

This paper develops a symbolic coding for the geodesic flow on Hecke triangle surfaces, linking it to continued fraction expansions and providing explicit return times, enhancing understanding of the flow's dynamics.

Contribution

It introduces an explicit cross section and a simplified return map for the geodesic flow on Hecke surfaces, connecting it to continued fractions.

Findings

Explicit cross section for geodesic flow constructed

Return map factors through a simple continued fraction map

Explicit formulas for first return times derived

Abstract

In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.