Center manifolds for difference equations—smooth parameter dependence

For sufficiently small C1C1 perturbations of (nonautonomous) linear difference equations with a nonuniform exponential trichotomy, we establish the existence of center manifolds with the optimal C1C1 regularity. We also consider the case of parameter-dependent perturbations and we obtain the C1C1 dependence of the center manifolds on the parameter. In addition, we consider arbitrary growth rates with the usual exponential estimates of the form ect in the notion of exponential trichotomy replaced by ecρ(t) where ρρ is now an arbitrary function. The proof of the regularity, both of the center manifold and of its dependence, on the parameter is based on the fiber contraction principle. The most technical part of the argument concerns the continuity of the fiber contraction that essentially needs a direct argument.