Function theory, Dynamics and Ergodic theory via Thurston theory

TL;DR

This paper explores the interplay between function theory on Teichmüller space, the dynamics of mapping class groups, and ergodic actions, integrating Thurston's and Sullivan's theories to advance understanding in these interconnected areas.

Contribution

It introduces a novel framework connecting Thurston's theory with ergodic and dynamical properties of mapping class groups.

Findings

New insights into the dynamics of subgroups of the mapping class group.

Connections established between Teichmüller theory and ergodic actions.

Enhanced understanding of hyperbolic space at infinity.

Abstract

In this paper, we discuss function theory on Teichm\"uller space through Thurston's theory, as well as the dynamics of subgroups of the mapping class group of a surface, with reference to Sullivan's theory on the ergodic actions of discrete subgroups of the isometry group of hyperbolic space at infinity.