Iteration of the coupled map lattice construction

We define a class 3 of topological dynamical systems, which are left invariant by coupled map lattice constructions. ‘fhis class .F has the property that if the coupling of the systems is sufficiently weak and (M, f) contains a hyperbolic set, ‘hen the new system $( M, f) obtained by the coupled map lattice construction has a hyperbolic set too. The coupled map attice construction, map 4, can be iterated and leads to dynamical systems (M,, fn) with a hierarchical diffusion structure. ‘We obtain examples where shifts are embedded in all scales of the limiting system (Mm, foe) E F.