The Weil-Petersson metric is a Riemannian metric on the Teichmueller space of a Riemann surface. It is invariant under the action of the mapping class group and descends to a Riemannian metric with volume on the moduli space.

The Weil-Peterson metric has negative sectional curvatures but is incomplete. The curvature and its derivatives blow up as one approaches the boundary of Teichmueller space. The effect of negative curvature on the behaviour of the geodesic flow is well understood. In particular Hopf and Anosov showed that the geodesic flow of a compact manifold with negative curvatures is ergodic. Their results extend to the Weil-Petersson geodesic flow, but the incompleteness of the metric creates considerable additional difficulties.

The lecture will attempt  to outline the arguments of Hopf and Anosov and to indicate the additional ideas needed to apply them to the Weil-Petersson  geodesic flow.