Axiom A flows and Anosov representations

In a joint work with B. Delarue and A. Sanders, we construct Axiom A flows on locally homogeneous manifolds associated to Anosov representations into a real semi-simple Lie group G, generalizing the geodesic flow of a hyperbolic manifold in the case of G = SO(n,1). We will see examples for which these flows can be interpreted as geodesic flows on pseudo-Riemannian symmetric spaces, such as G = SO(2,2) (related to anti-de Sitter geometry, studied in joint work with B. Delarue and C. Guillarmou), and others for which there is no such interpretation.