Synchronization versus stability of the invariant distribution for a class of globally coupled maps

We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength $\varepsilon$. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength $\varepsilon$ is Lipschitz continuous in the BV norm. When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state.