  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.
