Globally coupled Anosov diffeomorphisms: Statistical properties

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_{\epsilon }$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\epsilon \mapsto h_{\epsilon }$ is Lipschitz continuous.