  We show that under certain simple assumptions on the topology (structure) of
networks of strongly interacting chaotic elements a phenomenon of long range
action takes place, namely that the asymptotic (as time goes to infinity)
dynamics of an arbitrary large network is completely determined by its boundary
conditions. This phenomenon takes place under very mild and robust assumptions
on local dynamics with short range interactions. However, we show that it is
unstable with respect to arbitrarily weak local random perturbations.
