Horospherical invariant measures and a rank dichotomy for Anosov groups

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.

arXiv comment33 pages (to appear in Journal of Modern Dynamics)