ITERATION OF THE COUPLED MAP LATTICE CONSTRUCTION

We define a class F of topological dynamical systems, which are left invariant by coupled map lattice constructions. This class F has the property that if the coupling of the systems is sufficiently weak and (M, f) contains a hyperbolic set, then the new system phi(M, f) obtained by the coupled map lattice construction has a hyperbolic set too. The coupled map lattice construction, map phi, can be iterated and leads to dynamical systems (M(n), f(n)) with a hierarchical diffusion structure. We obtain examples where shifts are embedded in all scales of the limiting system (M(infinity), f(infinity)) is an element of F.