Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature

TL;DR

This paper develops a new family of arithmetic cross-sections for geodesic flow on compact negatively curved surfaces, using boundary maps with finite attractors, extending and simplifying previous results, and computing their entropy.

Contribution

It introduces a novel construction of cross-sections based on boundary maps with the short cycle property, extending Adler and Flatto's earlier work.

Findings

Boundary maps with the short cycle property have finite attractors.

The geodesic flow can be represented as a special flow over a symbolic system.

The measure-theoretic entropy of boundary maps is computed.

Abstract

We describe a family of arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature based on the study of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups and their natural extensions, introduced in Katok-Ugarcovici's 2017 paper. If the boundary map satisfies the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure, and the associated arithmetic cross-section is parametrized by the attractor. This construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. In special cases where the "cycle ends" are discontinuity points of the boundary maps, the resulting symbolic system is sofic. Thus we extend and in some ways simplify several results of Adler-Flatto's 1991 paper. We also compute the measure-theoretic entropy of the boundary maps.