Hölder Grobman-Hartman linearization

Luis Barreira; Claudia Valls

doi:10.3934/dcds.2007.18.187

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2007, Volume 18, Issue 1: 187-197. Doi: 10.3934/dcds.2007.18.187

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Hölder Grobman-Hartman linearization

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: January 31, 2006

Revised: November 30, 2006

Published: December 2006

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Abstract

We prove that the conjugacies in the Grobman-Hartman theorem are always Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. We also consider the case of hyperbolic trajectories of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. All the results are obtained in Banach spaces. It is common knowledge that some authors claimed that the Hölder regularity of the conjugacies is well known, however without providing any reference. In fact, to the best of our knowledge, the proof appears nowhere in the published literature.

Keywords:

Grobman--Hartman theorem,
Hölder regularity.

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

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