  By constructing a non-empty domain of discontinuity in a suitable homogeneous
space, we prove that every torsion-free projective Anosov subgroup is the
monodromy group of a locally homogeneous contact Axiom A dynamical system with
a unique basic hyperbolic set on which the flow is conjugate to the refraction
flow of Sambarino. Under the assumption of irreducibility, we utilize the work
of Stoyanov to establish spectral estimates for the associated complex Ruelle
transfer operators, and by way of corollary: exponential mixing, exponentially
decaying error term in the prime orbit theorem, and a spectral gap for the
Ruelle zeta function. With no irreducibility assumption, results of
Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions
with smooth weights, as well as the existence of a discrete spectrum of
Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to
space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds
of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for
strictly convex real projective manifolds.
