A report on an ergodic dichotomy

TL;DR

This paper explores the relationship between cocycles and measures in hyperbolic groups and representations, establishing a dichotomy in limit behaviors based on geometric parameters, with some cases still unresolved.

Contribution

It introduces a new dichotomy for Patterson-Sullivan measures in Anosov representations, linking measure support to geometric parameters of the representation.

Findings

For $| heta|\leq 2$, the $u$-conical limit points have total measure.

For $| heta|\geq 4$, the $u$-conical limit points have zero measure.

The case $| heta|=3$ remains unresolved.

Abstract

We establish (some directions) of a Ledrappier correspondence between H\"older cocycles, Patterson-Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson-Sullivan measures for $\theta$-Anosov representations over a local field and show that these are parametrized by the $\theta$-critical hypersurface of the representation. We use these Patterson-Sullivan measures to establish a dichotomy concerning directions in the interior of the $\theta$-limit cone of the representation in question: if $u$ is such a half-line, then the subset of $u$-conical limit points has either total-mass if $|\theta|\leq2$ or zero-mass if $|\theta|\geq4.$ The case $|\theta|=3$ remains unsettled.