Uniqueness of conformal measures and local mixing for Anosov groups

In the late seventies, Sullivan showed that for a convex cocompact subgroup $Γ$ of $\operatorname{SO}^\circ(n,1)$ with critical exponent $δ>0$, any $Γ$-conformal measure on $\partial \mathbb{H}^n$ 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 of a connected semisimple real algebraic group $G$ of rank at most $3$. We also obtain the local mixing for generalized BMS measures on $Γ\backslash G$ including Haar measures.

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