Measurable boundary maps and Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary

TL;DR

This paper develops a comprehensive theory for Patterson--Sullivan measures on the Furstenberg boundary for non-Borel Anosov groups, extending previous results and establishing new ergodic and rigidity properties.

Contribution

It introduces a new sufficient condition for measurable boundary maps applicable to various groups, and extends measure theory to broader Anosov conditions.

Findings

Existence and uniqueness of measures on the Furstenberg boundary under arbitrary Anosov conditions

Ergodicity of Bowen--Margulis--Sullivan measures on homogeneous spaces

Strict convexity of the critical exponent and entropy rigidity results

Abstract

In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold associated to the Anosov condition, which coincides with the Furstenberg boundary only under the strongest Anosov condition, Borel Anosov. We establish existence, uniqueness, and ergodicity results for the measures on the Furstenberg boundary under arbitrary Anosov conditions; we show ergodicity of Bowen--Margulis--Sullivan measures on the homogeneous space; and we establish strict convexity results for the critical exponent associated to functionals on the entire Cartan subspace. Using this strict convexity, we establish an entropy rigidity result for Anosov groups with Lipschitz limit set. A key tool we develop is a new sufficient condition for the existence of a measurable boundary map associated to a Zariski dense representation. This result not only applies to Anosov groups, but also to transverse groups, mapping class groups, and discrete subgroups of the isometry groups of Gromov hyperbolic spaces.