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 the corresponding invariant manifolds.

Peer Reviewed