Non-concentration property of Patterson-Sullivan measures for Anosov subgroups

Abstract

Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_{\theta }$, we prove that any $\Gamma$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_{\theta }$. More generally, we prove that for a Zariski dense $\theta$-transverse subgroup $\Gamma <G$, any $(\Gamma ,\psi )$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_{\theta }$, provided the $\psi$-Poincaré series of $\Gamma$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups.

arXiv comment10 pages, Final version, To appear in Ergodic Theory Dynam. Systems