Characterization of nonuniform exponential trichotomies for flows

For an evolution family, we characterize completely the notion of a nonuniform exponential trichotomy in terms of admissibility properties. As a nontrivial application, we establish the robustness of the notion in a very simple manner. We also obtain corresponding results for a strong nonuniform exponential trichotomy. We emphasize that both notions are ubiquitous in the context of ergodic theory. Moreover, we develop a corresponding theory for the notion of a nonuniformly partially hyperbolic set, in which case one considers simultaneously a collection of trajectories instead of a single one. In particular, this required first developing an appropriate theory for nonuniformly hyperbolic sets, which is of independent interest.