2004/07/20 by Alexander Gorodnik, Gorodnik, Alexander, Barak Weiss +1
Mathematics · #11H55 #22E40 #37A17 #57S30 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/0407345
openalex publication_date 2004/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a lattice Γin a locally compact group G and a closed subgroup H of G, one has a natural action of Γon the homogeneous space V=H\G. For an increasing family of finite subsets ΓT: T>0, a dense orbit vΓ, v∈ V, and compactly supported function ϕon V, we consider the sums Sϕ,v(T)=∑γ∈ ΓT ϕ(v γ). Understanding the asymptotic behavior of Sϕ,v(T) is a delicate problem which has only been considered for certain very special choices of H, G and ΓT. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have Sϕ,v(T) ∼ ∫GT ϕ(vg) dg, where GT=g∈ G:||g||