Locally homogeneous Axiom A flows II: geometric structures for Anosov subgroups

Given a non-compact semisimple real Lie group G and an Anosov subgroup Γ, we utilize the correspondence between R-valued additive characters on Levi subgroups L of G and R-affine homogeneous line bundles over G/L to systematically construct families of non-empty domains of proper discontinuity for the Γ-action. If Γ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the R-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of G.