Stable manifolds with optimal regularity for difference equations

Luis Barreira; Claudia Valls

doi:10.3934/dcds.2012.32.1537

Article Contents

2012, Volume 32, Issue 5: 1537-1555. Doi: 10.3934/dcds.2012.32.1537

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Stable manifolds with optimal regularity for difference equations

Luis Barreira^1, and
Claudia Valls^2,

1.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa
2.
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received: December 2010

Revised: April 2011

Published: May 2012

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Abstract

We obtain stable invariant manifolds with optimal $C^k$ regularity for a nonautonomous dynamics with discrete time. The dynamics is obtained from a sufficiently small perturbation of a nonuniform exponential dichotomy, which includes the notion of (uniform) exponential dichotomy as a very special case. We emphasize that we do not require the dynamics to be of class $C^{k+\epsilon}$, in strong contrast to former results in the context of nonuniform hyperbolicity. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, our method also allows linear perturbations, and thus the results readily apply to the robustness problem of nonuniform exponential dichotomies.

Keywords:

Difference equations,
stable manifolds.

Mathematics Subject Classification: Primary: 37D10, 37D25.

Citation:

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References

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