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arXiv · 2106.02635

Horospherical invariant measures and a rank dichotomy for Anosov groups

Or Landesberg·Minju Lee·Elon Lindenstrauss·Hee Oh
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Abstract

Let $G=\prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of rank one and $\Gamma$ an Anosov subgroup of $G$ with respect to a minimal parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber, let $\mathcal R_v\subset\Gamma\backslash G$ denote the Borel subset of all points with recurrent $\exp (\mathbb R_+ v)$-orbits. For a maximal horospherical subgroup $N$ of $G$, we show that the $N$-action on ${\mathcal R}_v$ is uniquely ergodic if $r={rank}(G)\le 3$ and $v$ belongs to the interior of the limit cone of $\Gamma$, and that there exists no $N$-invariant {Radon} measure on $\mathcal R_v$ otherwise.

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Or Landesberg, Minju Lee, Elon Lindenstrauss, Hee Oh. 2021-06-04. Horospherical invariant measures and a rank dichotomy for Anosov groups. https://arxiv.org/abs/2106.02635

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