Uniqueness of conformal measures and local mixing for Anosov groups

TL;DR

This paper extends Sullivan's uniqueness theorem for conformal measures to Zariski dense Anosov subgroups of certain semisimple groups and establishes local mixing for associated measures.

Contribution

It generalizes Sullivan's classical results to a broader class of groups and proves local mixing for generalized BMS measures on these spaces.

Findings

Uniqueness of conformal measures for Zariski dense Anosov subgroups.

Establishment of local mixing properties for generalized BMS measures.

Extension of classical hyperbolic geometry results to higher rank semisimple groups.

Abstract

In the late seventies, Sullivan showed that for a convex cocompact subgroup $\Gamma$ of $\operatorname{SO}^\circ(n,1)$ with critical exponent $\delta>0$, any $\Gamma$-conformal measure on $\partial \mathbb{H}^n$ of dimension $\delta$ is necessarily supported on the limit set $\Lambda$ and that the conformal measure of dimension $\delta$ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$ of rank at most $3$. We also obtain the local mixing for generalized BMS measures on $\Gamma\backslash G$ including Haar measures.