Horospherical invariant measures and a rank dichotomy for Anosov groups

Abstract

Let $ G = \prod_{i = 1}^{\mathsf r} G_i $ be a product of simple real algebraic groups of rank one and $ \Gamma $ an Anosov subgroup of $ G $ with respect to a minimal parabolic subgroup. For each $ \mathsf v $ in the interior of a positive Weyl chamber, let $ \mathscr R_ \mathsf v\subset \Gamma\backslash G $ denote the Borel subset of all points with recurrent $ \exp (\mathbb R_+ \mathsf v) $-orbits. For a maximal horospherical subgroup $ N $ of $ G $, we show that the $ N $-action on $ {\mathscr R}_ \mathsf v $ is uniquely ergodic if $ \mathsf r = \operatorname{rank}(G)\le 3 $ and $ \mathsf v $ belongs to the interior of the limit cone of $ \Gamma $, and that there exists no $ N $-invariant Radon measure on $ \mathscr R_ \mathsf v $ otherwise.

Locations

• arXiv (Cornell University) - View - PDF
• Journal of Modern Dynamics - View - PDF