Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity

TL;DR

This paper introduces Patterson--Sullivan systems, generalizes boundary rigidity theorems, and applies these results to study measures, random walks, and entropy rigidity in hyperbolic and higher rank spaces.

Contribution

It generalizes Tukia's boundary rigidity theorem to Patterson--Sullivan systems and applies this to various groups and spaces, including hyperbolic and higher rank symmetric spaces.

Findings

Proved a generalization of Tukia's boundary rigidity theorem.

Applied the generalization to study singularity conjectures and measures.

Established versions of Tukia's theorem for diverse geometric groups and spaces.

Abstract

In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichm\"uller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets.