Mean field coupled dynamical systems: Bifurcations and phase transitions

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled expanding maps to globally coupled Axiom A diffeomorphisms. We analyze an explicit example consisting of globally coupled Anosov diffeomorphisms. For such an example, we classify all the invariant measures as the coupling strength varies; we show which invariant measures are physical, and we prove that the existence of multiple invariant physical measures is an infinite dimensional phenomenon, i.e., the model exhibits phase transitions in the sense of statistical mechanics.