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

TL;DR

This paper constructs and analyzes geometric structures and dynamical systems related to Anosov subgroups in semisimple Lie groups, revealing new connections between hyperbolic flows, foliations, and Lie algebra Poisson structures.

Contribution

It introduces a systematic method to build domains of proper discontinuity and links Axiom A flows with geometric structures like pseudo-Riemannian metrics and contact forms.

Findings

Existence of non-empty domains of proper discontinuity for Anosov subgroup actions

Construction of Axiom A dynamical systems with stable/unstable foliations

Connection between additive characters, geometric structures, and Poisson brackets

Abstract

Given a non-compact semisimple real Lie group $G$ and an Anosov subgroup $\Gamma$, we utilize the correspondence between $\mathbb R$-valued additive characters on Levi subgroups $L$ of $G$ and $\mathbb R$-affine homogeneous line bundles over $G/L$ to systematically construct families of non-empty domains of proper discontinuity for the $\Gamma$-action. If $\Gamma$ 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 $\mathbb R$-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 $G$.