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Conjugacies between linear and nonlinear non-uniform contractions

Published online by Cambridge University Press:  01 February 2008

LUIS BARREIRA  and
CLAUDIA VALLS
LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (email: [email protected], [email protected])
CLAUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (email: [email protected], [email protected])
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Abstract

We construct conjugacies between linear and nonlinear non-uniform exponential contractions with discrete time. We also consider the general case of a non-autonomous dynamics defined by a sequence of maps. The results are obtained by considering both linear and nonlinear perturbations of the dynamics xm+1=Amxm defined by a sequence of linear operators Am. In the case of conjugacies between linear contractions we describe them explicitly. All the conjugacies are locally Hölder, and in fact are locally Lipschitz outside the origin. We also construct conjugacies between linear and nonlinear non-uniform exponential dichotomies, building on the arguments for contractions. All the results are obtained in Banach spaces.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 1 , February 2008 , pp. 1 - 19
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Barreira, L. and Valls, C.. A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics. J. Differential Equations 228 (2006), 285–310.CrossRefGoogle Scholar
[2]Grobman, D.. Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1959), 880–881.Google Scholar
[3]Grobman, D.. Topological classification of neighborhoods of a singularity in n-space. Mat. Sb. (N.S.) 56(98) (1962), 77–94.Google Scholar
[4]Hartman, P.. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960), 610–620.CrossRefGoogle Scholar
[5]Hartman, P.. On the local linearization of differential equations. Proc. Amer. Math. Soc. 14 (1963), 568–573.CrossRefGoogle Scholar
[6]Palis, J.. On the local structure of hyperbolic points in Banach spaces. An. Acad. Brasil. Ciênc. 40 (1968), 263–266.Google Scholar
[7]Pugh, C.. On a theorem of P. Hartman. Amer. J. Math. 91 (1969), 363–367.CrossRefGoogle Scholar
[8]Sternberg, S.. Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809–824.Google Scholar
[9]Sternberg, S.. On the structure of local homeomorphisms of euclidean n-space. II. Amer. J. Math. 80 (1958), 623–631.CrossRefGoogle Scholar