Uniqueness of conformal measures and local mixing for Anosov groups

Abstract

In the late seventies, Sullivan showed that for a convex cocompact subgroup $\Gamma$ of ${\mathrm{SO}}^{\circ }(n,1)$ 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.

17 pages, To appear in Michigan Mathematical Journal (Prasad volume)