Uniqueness of Conformal Measures and Local Mixing for Anosov Groups

Abstract

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