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.
Categories: math.DS