Characterization of strong exponential dichotomies

Abstract

We introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility of bounded solutions. The latter refers to the existence of (unique) bounded solutions for any bounded perturbation of the original dynamics. We consider the general case of a nonautonomous dynamics defined by a sequence of linear operators. As a nontrivial application, we establish the robustness of nonuniform exponential dichotomies as well as of strong nonuniform exponential dichotomies, which corresponds to the persistence of these notions under sufficiently small linear perturbations. The relevance of the results stems from the ubiquity of this type of exponential behavior in the context of ergodic theory: for almost all trajectories with nonzero Lyapunov exponents of a measure-preserving diffeomorphism, the derivative cocycle admits a nonuniform exponential dichotomy and in fact a strong nonuniform exponential dichotomy.