Expansive factors for geodesic flows of compact manifolds without conjugate points and with visibility universal covering

TL;DR

This paper investigates the dynamics of geodesic flows on compact manifolds without conjugate points, establishing the existence of an expansive, topologically mixing factor with a unique measure of maximal entropy under certain conditions.

Contribution

It introduces a new expansive factor for geodesic flows on such manifolds and proves the uniqueness of the measure of maximal entropy assuming entropy-gap and additional geometric conditions.

Findings

Existence of a topologically mixing, expansive factor for the geodesic flow.

Uniqueness of the measure of maximal entropy under entropy-gap.

Green bundles are uniquely integrable and tangent to horospherical foliations.

Abstract

Let $(M,g)$ be a compact manifold without conjugate points and with visibility universal covering. We show that its geodesic flow has a time-preserving expansive factor which is topologically mixing and has a local product structure. As an application, assuming further the so-called entropy-gap we prove the uniqueness of the measure of maximal entropy for the geodesic flow. For the other results we restrict our setting assuming furthermore the continuity of Green bundles and the existence of a hyperbolic closed geodesic. In this new context, we deduce that Green bundles are uniquely integrable and are tangent to the smooth leaves of the horospherical foliations. Moreover, we prove the above expansive factor acts on a compact topological manifold and its geodesic flow has a unique measure of maximal entropy which has full support.