Invariant measures for Deroin-Tholozan representations

TL;DR

This paper classifies invariant probability measures on character varieties of Deroin-Tholozan representations, showing they are either finite orbit measures or Liouville measures, using measure disintegration techniques.

Contribution

It provides a complete classification of invariant measures on these character varieties, revealing their structure and ergodic properties.

Findings

Invariant measures are either finite orbit counting measures or Liouville measures.

Ergodic measures are classified explicitly.

The approach uses measure disintegration along Lagrangian tori fibrations.

Abstract

We classify mapping class group invariant probability measures on the character varieties of Deroin-Tholozan representations, namely the compact components of relative $\mathrm{PSL}_2\mathbb{R}$-character varieties. We prove that an ergodic measure is either the counting measure on a finite orbit or agrees with the Liouville measure induced by the Goldman symplectic form. Our approach is based on measure disintegration along transverse Lagrangian tori fibrations.