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.
Keywords:
exponential trichotomy, robustness, partially hyperbolic set
Citation:
Barreira L., Dragičević D., Valls C., Admissibility and Nonuniform Exponential Trichotomies, Regular and Chaotic Dynamics,
2015, Volume 20, Number 1,
pp. 49-62