A Grobman–Hartman theorem for general nonuniform exponential dichotomies

For a nonautonomous dynamics with discrete time given by a sequence of linear operators Am, we establish a version of the Grobman–Hartman theorem in Banach spaces for a very general nonuniformly hyperbolic dynamics. More precisely, we consider a sequence of linear operators whose products exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(n), determined by a sequence ρ(n). For all sufficiently small Lipschitz perturbations Am + fm we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators Am.We also show that all conjugacies are Hölder continuous.We note that the usual exponential behavior is included as a very special case when ρ(n) = n, but many other asymptotic behaviors are included such as the polynomial asymptotic behavior when ρ(n) = log n. © 2009 Elsevier Inc. All rights reserved.