Smooth linearization under nonuniform hyperbolicity

Abstract

For any sufficiently small perturbation of a tempered exponential dichotomy, we obtain appropriate versions of the Grobman–Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker–Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method.