Matteo Tanzi (King's College London)
Uniformly Expanding Coupled Maps: Self-Consistent Transfer Operators and Propagation of Chaos
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.