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First-return maps of Birkhoff sections of the geodesic flow

Théo Marty

Algebraic & Geometric Topology 22 (2022) 2355–2394
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.

Keywords
geodesic flow, Birkhoff section, first-return map, fibered face
Mathematical Subject Classification
Primary: 57M50
Secondary: 37D20
References
Publication
Received: 29 September 2020
Revised: 10 March 2021
Accepted: 8 May 2021
Published: 25 October 2022
Authors
Théo Marty
Institut Fourier
Université Grenoble Alpes
Gières
France