Optimal regularity of stable manifolds of nonuniformly hyperbolic dynamics

Abstract

We establish the existence of smooth invariant stable manifolds for differential equations $u'=A(t)u+f(t,u)$ obtained from sufficiently small perturbations of a {\it nonuniform} exponential dichotomy for the linear equation $u'=A(t)u$. One of the main advantages of our work is that the results are optimal, in the sense that the invariant manifolds are of class $C^k$ if the vector field is of class $C^k$. To the best of our knowledge, in the nonuniform setting this is the first general optimal result (for a large family of perturbations and not for some specific perturbations). Furthermore, in contrast to some former works, we do not require a strong nonuniform exponential behavior (we note that contrarily to what happens for autonomous equations, in the nonautonomous case a nonuniform exponential dichotomy need not be strong). The novelty of our proofs, in this setting, is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, we can also consider linear perturbations, and our results have thus immediate applications to the robustness of nonuniform exponential dichotomies.