Horospherical invariant measures and a rank dichotomy for Anosov groups

TL;DR

This paper investigates invariant measures for horospherical actions on Anosov groups, revealing a rank-dependent dichotomy in ergodic behavior and measure existence.

Contribution

It establishes a rank-based dichotomy for the uniqueness of invariant measures under horospherical actions on Anosov groups, extending understanding of dynamical systems in higher rank.

Findings

Unique ergodicity for ranks ≤ 3 when v is in the limit cone

Non-existence of N-invariant Radon measures for ranks > 3

Characterization of recurrent orbit subsets in 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.