Local Mixing of One-Parameter Diagonal Flows on Anosov Homogeneous Spaces

Abstract

Abstract Let $G$ be a connected semisimple real algebraic group and $\Gamma < G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp (t\mathsf {v}): t \in {\mathbb {R}}\}$ on $\Gamma \backslash G$ for any interior direction $\mathsf {v}$ of the limit cone of $\Gamma $ with respect to the Bowen–Margulis–Sullivan measure associated to $\mathsf {v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf {u}$ in some fixed subspace transverse to $\mathsf {v}$. We also obtain a uniform bound for the correlation function, which decays exponentially in $\|\mathsf {u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(\Gamma \backslash G)$ proved by Edwards–Lee–Oh.