Abstract
AbstractWe 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
$\vartheta $
-Anosov representations over a local field and show that these are parameterized by the
$\vartheta $
-critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the
$\vartheta $
-limit cone of the representation in question: if
${\mathsf {u}}$
is such a half-line, then the subset of
${\mathsf {u}}$
-conical limit points has either total mass if
$|\vartheta |\leq 2$
or zero mass if
$|\vartheta |\geq 4.$
The case
$|\vartheta |=3$
remains unsettled.
Funder
Agence Nationale de la Recherche
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics