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.