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 -Anosov representations over a local field and show that these are parameterized by the - critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the -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 | | 2 or zero mass if | | 4. The case | |=3 remains unsettled.