Genus one Birkhoff sections for geodesic flows

TL;DR

This paper constructs explicit genus one Birkhoff sections for geodesic flows on hyperbolic 2-orbifolds with 3 or 4 singular points, providing detailed return maps.

Contribution

It introduces explicit genus one Birkhoff sections for geodesic flows on certain hyperbolic orbifolds, expanding understanding of their dynamical structure.

Findings

Existence of explicit genus one Birkhoff sections for specified orbifolds

Determination of the associated first return maps

Enhanced understanding of geodesic flow dynamics on hyperbolic orbifolds

Abstract

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.