Effective Mixing and Counting in Bruhat-Tits Trees

Abstract

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

28 pages, 7 figures