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. then study measures $\vartheta $ -Anosov representations over local field and show that these are parameterized by the - critical hypersurface of representation. use to dichotomy concerning in interior -limit cone representation question: if ${\mathsf {u}}$ is such half-line, subset conical limit points has either total mass $|\vartheta |\leq 2$ or zero |\geq 4.$ The case |=3$ remains unsettled.