Gerhard Keller
An ergodic theoretic approach to mean field coupled maps
(77K, LaTeX2e)
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 i.i.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 $C^3$-expanding maps of the circle (extending slightly a result of
J\"arvenp\"a\"a) nor for mixing tent maps for which the critical orbit
finally hits a fixed point (making rigorous a result of Chawanya and
Morita).