Tempered exponential dichotomies: admissibility and stability under perturbations

We give a characterization of the notion of a tempered exponential dichotomy on a Banach space in terms of an admissibility property. We note that for a linear cocycle over a measure-preserving transformation satisfying a certain integrability assumption, it follows from the multiplicative ergodic theorem that the dynamics admits a tempered exponential dichotomy if and only if all Lyapunov exponents are non-zero almost everywhere. As a consequence of our approach, we give a new proof of the robustness property of the notion of a tempered exponential dichotomy under sufficiently small linear perturbations and we establish a version of the Grobman–Hartman theorem yielding the existence of topological conjugacies between a linear dynamics with a tempered exponential dichotomy and any sufficiently small nonlinear perturbation. In addition, we show that the conjugacy maps vary continuously with the perturbation.