Topological conjugacies and behavior at infinity

Luis Barreira; Claudia Valls

doi:10.3934/cpaa.2014.13.687

Article Contents

2014, Volume 13, Issue 2: 687-701. Doi: 10.3934/cpaa.2014.13.687

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Topological conjugacies and behavior at infinity

Luis Barreira^1, and
Claudia Valls^2,

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

Received: February 28, 2013

Revised: May 31, 2013

Published: February 2014

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Abstract

We obtain a version of the Grobman--Hartman theorem in Banach spaces for perturbations of a nonuniform exponential contraction, both for discrete and continuous time. More precisely, we consider the general case of an exponential contraction with an arbitrary nonuniform part and obtained from a nonautonomous dynamics, and we establish the existence of Hölder continuous conjugacies between an exponential contraction and any sufficiently small perturbation. As a nontrivial application, we describe the asymptotic behavior of the topological conjugacies in terms of the perturbations: namely, we show that for perturbations growing in a certain controlled manner the conjugacies approach zero at infinity and that when the perturbations decay exponentially at infinity the conjugacies have the same exponential behavior.

Keywords:

Exponential contractions,
topological conjugacies.

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

Citation:

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References

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