First-return maps of Birkhoff sections of the geodesic flow

Abstract

Given a hyperbolic orientable surface and a collection of periodic geodesics on it, we study the Birkhoff sections for the geodesic flow on the unit bundle to the surface bounded by this collection. The first-return maps on these Birkhoff sections are pseudo-Anosov maps, which we explicitly compute. We show that there is a canonical identification of all those Birkhoff sections, and that the first-return maps induced by the flow can all be expressed as a composition of negative Dehn twists along a family of explicit curves; only the order depends on the choice of a particular Birkhoff section.