Lecture notes on the dynamics of the Weil-Petersson flow

TL;DR

This paper discusses the dynamics of the Weil-Petersson flow on moduli spaces of Riemann surfaces, focusing on aspects related to ergodicity proven by Burns, Masur, and Wilkinson.

Contribution

It provides lecture notes that explain key concepts and recent results on the ergodicity of the Weil-Petersson geodesic flow within Teichmüller theory.

Findings

Analysis of the ergodicity of WP flow

Connection to moduli space dynamics

Insights into Teichmüller theory

Abstract

This text grew out of some lecture notes prepared by the author in the occasion of a series of three lectures during the workshop "Young mathematicians in dynamical systems" organized by Francoise Dal'bo, Louis Funar, Boris Hasselblatt and Barbara Schapira in November 2013 at CIRM, Marseille, France. The three lectures at the origin of this text were part of a minicourse by Keith Burns, Boris Hasselblatt and the author around the recent theorem of Burns-Masur-Wilkinson on the ergodicity of the Weil-Petersson (WP) geodesic flow. The goal of these notes is the same of the author's lectures: we cover some of the aspects related to moduli spaces of Riemann surfaces (and Teichmueller theory) in the Burns-Masur-Wilkinson's proof of the ergodicity of WP flow.