Uniqueness of conformal measures and local mixing for Anosov groups

Sam Edwards; Minju Lee; Hee Oh

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arXiv · 2111.12752

Uniqueness of conformal measures and local mixing for Anosov groups

Sam Edwards·Minju Lee·Hee Oh

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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.

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Sam Edwards, Minju Lee, Hee Oh. 2021-11-24. Uniqueness of conformal measures and local mixing for Anosov groups. https://arxiv.org/abs/2111.12752

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