Anosov groups: local mixing, counting and equidistribution
Samuel C. Edwards, Minju Lee, Hee Oh
Abstract
Let G be a connected semisimple real algebraic group, and Γ<G a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients ⟨(exptv). f1,f2⟩ in L2(Γ∖G) as t→∞ for any f1,f2Cc(Γ∖G) and any vector v in them interior of the limit cone of Γ. These asymptotics involve higher-rank analogues of Burger–Roblin measures, which are introduced in this paper. As an application, for any affine symmetric subgroup Hof G, we obtain a bisector counting result for Γ–orbits with respect to the corresponding generalized Cartan decomposition of G. Moreover, we obtain analogues of the results of Duke, Rudnick and Sarnak as well as Eskin and McMullen for counting discrete Γ–orbits in affine symmetric spaces H∖G.