On the local structure of ω-limit sets of maps

Let X be a Banach space and let F : X → X be C¹, F(0) = 0. It is proved that, under certain conditions, the ω-limit set of a trajectory contains a point of the unstable manifold of 0 different from 0 as soon as it contains 0. The conditions on F involve the spectrum of F¹(0) (implying the existence of stable, unstable, center-unstable and center manifolds of 0) and the dynamics of F on the center manifold of 0. In addition, it is assumed that either the center-unstable space of 0 is finite dimensional or the trajectory is relatively compact. In a number of particular cases this result allows to prove convergence of trajectories.