Dynamical systems on lattices with decaying interaction I: A functional analysis framework

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 theorem 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 to 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 by Fontich et al. (2011) [10].

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

“…As emphasized in [FdlLM11a], ℓ ∞ (Z N ) has a very complicated dual space which cannot be identified with a space of sequences since there is no Riesz-representation theorem. As a consequence, we have that the matrix elements of an operator do not characterize the operator and, relatedly, the differential of a map is not represented by its partial derivatives.…”

“…We will develop some technology that allows to verify this assumption rather comfortably in the cases of interest. A much more thorough treatment can be found in [FdlLM11a].…”

“…The following Lemma has a simple proof that can be found in [FdlLM11a]. The most subtle point is that we need to verify that the linear operators are given by the matrix.…”

“…In [JdlL00,FdlLM11a], one can find definitions for Hölder spaces, which is useful for other applications (such as thermodynamic formalism).…”

Dynamical systems on lattices with decaying interaction II: Hyperbolic sets and their invariant manifolds

“…Here we apply the general theory developed in Fontich et al (2011) [3] to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map.…”