Abstract:
For a nonautonomous dynamics defined by a sequence of linear operators acting on a Banach space, we show that the notion of a nonuniform exponential trichotomy can be completely characterized in terms of admissibility properties. This refers to the existence of bounded solutions under any bounded time-dependent perturbation of certain homotheties of the original dynamics. We also consider the more restrictive notion of a strong nonuniform exponential trichotomy and again we give a characterization in terms of admissibility properties. We emphasize that both notions are ubiquitous in the context of ergodic theory. As a nontrivial application, we show in a simple manner that the two notions of trichotomy persist under sufficiently small linear perturbations. Finally, we obtain a corresponding characterization of nonuniformly partially hyperbolic sets.
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