  In this paper we consider lattice systems coupled by local interactions. We
prove invariant manifold theorems for whiskered tori (we recall that whiskered
tori are quasi-periodic solutions with exponentially contracting and expanding
directions in the linearized system). The invariant manifolds we construct
generalize the usual (strong) (un) stable manifolds and allow us to consider
also non-resonant manifolds. We show that if the whiskered tori are localized
near a collection of specific sites, then so are the invariant manifolds.
  We recall that the existence of localized whiskered tori has recently been
proven for symplectic maps and flows in Fontich et. al. (submitted), but our
results do not need that the systems are symplectic. For simplicity we will
present first the main results for maps, but we will show tha the result for
maps imply the results for flows. It is also true that the results for flows
can be proved directly following the same ideas.
