Abstract. For nonautonomous linear difference equations in a Banach space and admitting a very general type of dichotomy, we show:
i) the existence of global invariant manifolds for small Lipschitz perturbations of the linear equation;
ii) the existence of local invariant manifolds for locally Lipschitz perturbations of the linear equation;
iii) the persistence of the dichotomic behavior under small linear perturbations exactly with the same growth rates.

In the particular case of (μ,ν)-dichotomies, nonuniform exponential dichotomies and nonuniform polynomial dichotomies our results are new or improve previous known results.

This talk is based on joint work with C.M. Silva.