Mean field coupled dynamical systems: bifurcations of physical measures and phase transitions

After introducing the notion of mean-field coupled systems, I will review results on statistical properties of such systems in the infinite dimensional case. All these results pertain to the case of weak coupling strength  where exsistence of a unique physical measure is proved.  I will  then present a recent result/work in progress (joint with C. Liverani)  where we develop abstract framework  to study invariant measures of infinite dimensional coupled systems where the coupling strength is not necessarily weak. In particular, I will present an example  where the site dynamics is given by an Anosov map and show that in the finite dimensional case, the system always has a unique physical measure; however,  in the infinite dimensional case the system admits a unique physical measure in the weak coupling regime, while for moderate coupling strength it admits multiple physical measures.