Hölder Grobman-Hartman linearization

Pages: 187 - 197, Volume 18, Issue 1, May 2007

doi:10.3934/dcds.2007.18.187       Abstract        Full Text (174.6K)       Related Articles

Luís Barreira - Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (email)
Claudia Valls - Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (email)

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.

Received: February 2006;      Revised: December 2006;      Published: February 2007.