Abstract:

Recently, much progress has been made in the mathematical study of self-consistent transfer operators which describe the mean-field limit of globally coupled maps. Conditions for the existence of equilibrium measures (fixed points for the self-consistent transfer operator) have been given, and their stability under perturbations and linear response have been investigated. In this talk, I am going to describe some novel developments on dynamical systems made of N uniformly expanding coupled maps when N is finite but large. I will introduce self-consistent transfer operators that approximate the evolution of measures under the dynamics, and quantify this approximation explicitly with respect to N. Using this result, I will show that uniformly expanding coupled maps satisfy propagation of chaos when N tends to infinity, and I will characterize the absolutely continuous invariant measures for the finite dimensional system.