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

Abstract

This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) 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. We also describe how the decay properties of the maps are inherited by\ud the corresponding invariant manifolds.Peer ReviewedPostprint (published version

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

“…This proves that Φ is a C r diffeomorphism. Then, as is proved in [10] (see Lemma D.3 here), if δ is small enough,…”

“…The novelty is that, applying the formalism, we obtain that the invariant manifolds are decay. In the companion paper [10], we apply this framework to obtain a theory of hyperbolic sets with decay, in particular, their structural stability and the decay properties of their invariant manifolds.…”

“…We continue with Hölder functions with decay. We finish the section with a technical lemma, used later in the study of the lattice and in [10]. Section 3 is devoted to a stable manifold theorem for maps between ∞ spaces with decay, describing the decay properties of the manifolds.…”

“…Hence, we will rewrite the composition operators using sections instead of diffeomorphisms. Here we will describe the properties of some general operators of this kind between spaces of sections, to be particularized in the next paper [10].…”

“…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].…”