  We develop a bifurcation theory for infinite dimensional systems satisfying
abstract hypotheses that are 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 attractors. To illustrate
the range of applicability, 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 a purely infinite dimensional phenomenon, i.e., the model
exhibits phase transitions in the sense of statistical mechanics.
