Dynamical Systems on Lattices with Decaying Interaction I: a Functional Analysis Framework

We consider weakly coupled map lattices with a decaying interaction. That is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites. We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local. We use this framework to prove a stable manifold theorems and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper [FdlLM10].