Ergodicity of the mapping class group action on Deroin-Tholozan representations

TL;DR

This paper proves that the mapping class group action on the character variety of super-maximal representations of a punctured sphere's fundamental group into PSL(2,R) is ergodic, using symplectic methods.

Contribution

It establishes the ergodicity of the mapping class group action on a specific class of character varieties, extending previous understanding of their dynamical properties.

Findings

The action is ergodic on the character variety.

Symplectic methods effectively analyze the dynamics.

Provides new insights into the structure of super-maximal representations.

Abstract

This note investigates the dynamics of the mapping class group action on the character variety of super-maximal representations of the fundamental group of a punctured sphere into $\text{PSL}(2,\mathbb{R})$, discovered by Deroin and Tholozan. We apply symplectic methods developed by Goldman and Xia to prove that the action is ergodic.