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, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous.