E. Fontich, R. de la Llave, P.Martin
Dynamical systems on lattices with decaying interaction I:
A functional analysis framework.
(549K, PDF)
ABSTRACT. 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.