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

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