Topological dynamics of the Weil-Petersson geodesic flow

TL;DR

This paper proves topological transitivity and demonstrates complex dynamical features like horseshoes and positive entropy in the Weil-Petersson geodesic flow on moduli spaces, revealing rich chaotic behavior.

Contribution

It introduces a new approach to analyze the Weil-Petersson geodesic flow, establishing topological transitivity and complex dynamics in two-dimensional moduli spaces.

Findings

Proves topological transitivity of the flow

Shows existence of horseshoes and positive entropy

Demonstrates exponential growth of hyperbolic closed geodesics

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.