Dynamical properties of the Weil-Petersson metric

TL;DR

This paper studies the dynamics of the Weil-Petersson metric on Teichmüller space, showing the density of certain invariant measures and analyzing subgroup actions on the boundary.

Contribution

It introduces new results on the density of invariant measures supported on closed orbits for the Weil-Petersson flow.

Findings

Invariant measures supported on closed orbits are dense among all invariant measures.

Subgroups of the mapping class group act on the CAT(0)-boundary with rich dynamical properties.

The structure of the Weil-Petersson completion influences the dynamics of the flow.

Abstract

Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flow on moduli space which are supported on closed orbits is dense in the space of invariant Borel probability measures.