Effective Mixing and Counting in Bruhat-Tits Trees

Abstract

Let $\mathcal{T}$ be a locally finite tree, $Γ$ be a discrete subgroup of $\textrm{Aut}(\mathcal{T})$ and $\widetilde{F}$ be a $Γ$-invariant potential. Suppose that the length spectrum of $Γ$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map $ϕ\colon Γ\backslash S\mathcal{T}\to Γ\backslash S\mathcal{T}$ with respect to the measure $m_{Γ,F}^{ν^-,ν^+}$ under the assumption that $Γ$ is full and $(Γ,\widetilde{F})$ has weighted spectral gap property. We also obtain the effective formula for the number of $Γ$-orbits with weights in a Bruhat-Tits tree $\mathcal{T}$ of an algebraic group.

28 pages, 7 figures