Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

TL;DR

This paper extends Teichmüller dynamics to a flow on flat bundles over surfaces, analyzing its ergodic properties and mixing behavior, especially for compact Lie groups, linking it to the mapping class group action.

Contribution

It introduces a continuous flow on deformation spaces of surface representations, generalizing Teichmüller dynamics and exploring its ergodic and mixing properties.

Findings

Flow is strongly mixing for compact G.

Flow is ergodic for the Weil-Petersson geodesic lift.

Ergodic properties relate to the mapping class group action.

Abstract

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous lift of the Weil-Petersson geodesic local. flow.