Characterization of Nonuniform Contractions and Expansions with Growth Rates

Abstract

For a one-sided nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of (strong) nonuniform exponential contractions and (strong) nonuniform exponential expansions in terms of admissibility of certain pairs of sequence spaces. We allow asymptotic rates of the form \({e^{c\rho(n)}}\) determined by an arbitrary increasing sequence \({\rho(n)}\) that tends to infinity. For example, the usual exponential behavior with \({\rho(n) = n}\) is included as a very special case. As a nontrivial application of our work, we establish the robustness of (strong) nonuniform exponential contractions and (strong) nonuniform exponential expansions, that is, the persistence of those notions under sufficiently small linear perturbations.