Effective Mixing and Counting in Bruhat-Tits Trees

TL;DR

This paper proves exponential mixing of geodesic flows and provides effective counting formulas for orbits in Bruhat-Tits trees under certain spectral gap conditions, advancing understanding of dynamics on trees.

Contribution

It establishes exponential mixing and effective counting formulas for group actions on Bruhat-Tits trees with spectral gap assumptions, extending prior results to weighted settings.

Findings

Proved exponential mixing of geodesic translation map under non-arithmetic length spectrum.

Derived effective orbit counting formulas in Bruhat-Tits trees.

Established conditions for spectral gap and fullness of the group.

Abstract

Let $\mathcal{T}$ be a locally finite tree, $\Gamma$ be a discrete subgroup of $\textrm{Aut}(\mathcal{T})$ and $\widetilde{F}$ 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 $\phi\colon \Gamma\backslash S\mathcal{T}\to \Gamma\backslash S\mathcal{T}$ with respect to the measure $m_{\Gamma,F}^{\nu^-,\nu^+}$ under the assumption that $\Gamma$ is full and $(\Gamma,\widetilde{F})$ 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.