Let (M, g) be a compact connected \(C^{\infty }\) surface without conjugate points of genus greater than one and \(\phi _t\) be its geodesic flow. Using Patterson-Sullivan theory, Climenhaga-Knieper-War constructed a fully supported measure of maximal entropy for \(\phi _t\) supported in the expansive set, i.e., the set of vectors tangent to geodesics without strips. As a consequence, the expansive set is dense in the unit tangent bundle. This fact was known assuming no focal points as a consequence of a result of Coudène and Shapira. They showed that flat strips are periodic and hence form a set of zero measure in the unit tangent bundle. We give an alternative proof of the density of the expansive set. This proof is independent of Patterson-Sullivan measure and has a purely geometrical and dynamical flavor. We also show other consequences such as a sort of topological version of Eberlein’s characterization of Anosov geodesic flows.
