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

Let G be a connected semisimple real algebraic group and Γ < 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 (tv): t ∈ R\ on Γ G for any interior direction v of the limit cone of Γ with respect to the Bowen–Margulis–Sullivan measure associated to v. More generally, we allow a class of deviations to this flow along a direction u in some fixed subspace transverse to v. We also obtain a uniform bound for the correlation function, which decays exponentially in \|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 L²(Γ G) proved by Edwards–Lee–Oh.