A Grobman–Hartman theorem for general nonuniform exponential dichotomies

Abstract

For a nonautonomous dynamics with discrete time given by a sequence of linear operators A m , 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 e c ρ ( n ) , determined by a sequence ρ ( n ) . For all sufficiently small Lipschitz perturbations A m + f m we construct topological conjugacies between the dynamics defined by this sequence and the dynamics defined by the operators A m . 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 .