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

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.