Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one

TL;DR

This paper develops a geometric method to construct cross sections for geodesic flows on rank one locally symmetric orbifolds, leading to a simple symbolic dynamics and transfer operator, exemplified for a specific modular group.

Contribution

It introduces a new geometric construction of cross sections for geodesic flows on these orbifolds, simplifying the associated symbolic dynamics and transfer operator analysis.

Findings

Constructed explicit cross sections for the geodesic flow

Derived a simplified symbolic dynamics on the real line

Produced a transfer operator with a straightforward structure

Abstract

We present a strategy for a geometric construction of cross sections for the geodesic flow on locally symmetric orbifolds of rank one. We work it out in detail for $\Gamma\backslash H$, where $H$ is the upper half plane and $\Gamma=\PGamma_0(p)$, $p$ prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on $\R$. The transfer operator produced from this symbolic dynamics has a particularly simple structure.