Uniqueness of Conformal Measures and Local Mixing for Anosov Groups

Abstract

Abstract. In the late seventies, Sullivan showed that for a convex cocompact subgroup Γ of SO(n, 1) with critical exponent δ > 0, any Γ-conformal measure on ∂H 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 Γ\G including Haar measures.