Analytic invariant manifolds for sequences of diffeomorphisms

Abstract

We obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We emphasize that the maps that we consider are defined in a real Euclidean space, and thus, one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics. We establish the existence both of stable (and unstable) manifolds and of center manifolds. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds, but also for all their derivatives.

Analytic invariant manifolds for nonautonomous equations

“…Here we consider instead the case of a real analytic dynamics, mimicking to the possible extent our work in [2] in the case of discrete time. To the best of our knowledge, it exists nowhere in the literature an analytic stable manifold theorem for nonautonomous differential equations in the nonuniformly hyperbolic setting.…”