Topological dynamics of the Weil-Petersson geodesic flow

Abstract

We prove topological transitivity for the Weil Petersson geodesic flow for two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that exploits the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil Petersson geodesic flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite.