Robustness of Discrete Dynamics via Lyapunov Sequences

Abstract

For a nonautonomous dynamics with discrete time defined by a sequence of matrices, we give a complete characterization of nonuniform exponential contractions and nonuniform exponential dichotomies in terms of Lyapunov sequences. We note that these include as very special cases uniform exponential contractions and uniform exponential dichotomies. Due to the central role played by these properties in a substantial part of the theory of dynamical systems, in particular in connection with the study of stable and unstable invariant manifolds, it is important to have available optimal characterizations that are more amenable to check whether a given dynamics has such a property. We also obtain inverse theorems that give explicitly Lyapunov sequences for a given contraction or dichotomy. As a nontrivial application, we establish the robustness under sufficiently small linear perturbations both of nonuniform exponential contractions and nonuniform exponential dichotomies. We emphasize that when compared to former work, our proof of the robustness property is much simpler.