Geodesic flows modeled by expansive flows: Compact surfaces without conjugate points and continuous Green bundles

TL;DR

This paper investigates the structure of geodesic flows on compact surfaces without conjugate points, revealing a quotient manifold with an expansive flow and establishing uniqueness of the maximal entropy measure.

Contribution

It introduces a new quotient space with an expansive flow, linking geodesic flows to continuous Green bundles and analyzing their regularity properties.

Findings

The quotient space is a 3D compact manifold.

The induced flow is expansive and semi-conjugate to the geodesic flow.

The geodesic flow has a unique measure of maximal entropy.

Abstract

We study the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles. Identifying each strip of bi-asymptotic geodesics induces an equivalence relation on the unit tangent bundle. Its quotient space is shown to carry the structure of a 3-dimensional compact manifold. This manifold carries a canonically defined continuous flow which is expansive, time-preserving semi-conjugate to the geodesic flow, and has a local product structure. An essential step towards the proof of these properties is to study regularity properties of the horospherical foliations and to show that they are indeed tangent to the Green subbundles. As an application it is shown that the geodesic flow has a unique measure of maximal entropy.