An Ergodic Theoretic Approach to Mean Field Coupled Maps

Abstract

We study infinite systems of globally coupled maps with permutation invariant interaction as limits of large finite-dimensional systems. Because of the symmetry of the interaction the interesting invariant measures are the exchangeable ones. For infinite systems this means in view of de Finetti's theorem that we must look for time invariant measures within the class of mixtures of spatial Li.d. processes. If we consider only those invariant measures in that class as physically relevant which are weak limits of SRB-measures of the finite-dimensional approximations, we find for systems of piecewise expanding interval maps that the limit measures are in fact mixtures of absolutely continuous measures on the interval which have densities of uniformly bounded variation.

The law of large numbers is violated (in the sense of Kaneko) if a nontrivial mixture of i.i.d. processes can occur as a weak limit of finite-dimensional SRB-measures. We prove that this does neither happen for C3-expanding maps of the circle (extending slightly a result of Jiirvenpiiii) nor for mixing tent maps for which the critical orbit finally hits a fixed point (making rigorous a result of Chawanya and Morita).