Density of expansivity for geodesic flows of compact higher genus surfaces without conjugate points

TL;DR

This paper proves that in certain compact surfaces without conjugate points, geodesics without strips are densely distributed in the unit tangent bundle, extending previous results known under no focal points assumptions.

Contribution

It establishes the density of geodesics without strips on compact higher genus surfaces without conjugate points, generalizing prior work under no focal points conditions.

Findings

Geodesics without strips form a dense set in the unit tangent bundle.

Flat strips are periodic and have zero measure in the unit tangent bundle.

The result extends known properties from no focal points to conjugate points settings.

Abstract

Let $(M,g)$ be a compact connected $C^{\infty}$ surface without conjugate points of genus greater than one. We show that set of geodesics without strips forms a dense set of orbits in the unit tangent bundle. This fact was known assuming no focal points as a consequence of a result of Coud\`ene and Shapira. They showed that flat strips are periodic and hence form a set of zero measure in the unit tangent bundle.