A report on an ergodic dichotomy

Abstract

We establish (some directions of) a Ledrappier correspondence between Hölder cocycles, Patterson–Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson–Sullivan measures for ϑ-Anosov representations over a local field and show that these are parameterized by the ϑ-critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the ϑ-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 |ϑ| ≤ 2 or zero mass if |ϑ| ≥ 4. The case |ϑ| = 3 remains unsettled.