Anosov groups: local mixing, counting, and equidistribution

Abstract

Let $G$ be a connected semisimple real algebraic group, and $\Gamma <G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $⟨(\exp tv).f_1,f_2⟩$ in $L^2(\Gamma \backslash G)$ as $t\to \infty$ for any $f_1,f_2\in C_c(\Gamma \backslash G)$ and any vector $v$ in the interior of the limit cone of $\Gamma$. These asymptotics involve higher rank analogues of Burger-Roblin measures which are introduced in this paper. As an application, for any affine symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$.

52 pages, to appear in Geometry & Topology

References in corpus (1)

• Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces

Cited by in corpus (2)

• Temperedness of locally symmetric spaces: The product case
• Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank