### 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.

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.

#### Keywords

Nonuniform exponential dichotomies; stable manifolds

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