Invariant and stationary measures for the SL(2,R) action on Moduli space

TL;DR

This paper investigates the ergodic properties of SL(2,R) actions on moduli space, demonstrating that invariant measures under certain subgroups are supported on specific affine submanifolds, inspired by Ratner's work on unipotent flows.

Contribution

It establishes rigidity results for invariant measures under SL(2,R) actions on moduli space, extending ideas from homogeneous dynamics to this setting.

Findings

Invariant measures under upper triangular subgroup are supported on affine submanifolds

Results are inspired by Ratner's theorems on unipotent flows

Provides new rigidity properties for SL(2,R) actions on moduli space

Abstract

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner's seminal work.