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Theorem of Sternberg–Chen modulo the central manifold for Banach spaces

Published online by Cambridge University Press:  03 February 2009

VICTORIA RAYSKIN
VICTORIA RAYSKIN*
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva 84105, Israel (email: [email protected])
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Abstract

We consider C∞-diffeomorphisms on a Banach space with a fixed point 0 and linear part L. Suppose that these diffeomorphisms have C∞ non-contracting and non-expanding invariant manifolds, and formally conjugate along their intersection (the center). We prove that they admit local C∞ conjugation. In particular, subject to non-resonance conditions, there exists a local C∞ linearization of the diffeomorphisms. It also follows that a family of germs with a hyperbolic linear part admits a C∞ linearization, which has C∞ dependence on the parameter of the linearizing family. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces. The proofs of this paper are based on the technique, developed in the works of Belitskii [Funct. Anal. Appl.18 (1984), 238–239; Funct. Anal. Appl.8 (1974), 338–339].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1965 - 1978
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Belitskii, G.. Functional equations, and local conjugacy of mappings of class C∞. Math USSR-Sb. 20 (1973), 587–602.Google Scholar
[2]Belitskii, G.. The Sternberg theorem for a Banach space. Funct. Anal. Appl. 18 (1984), 238–239.CrossRefGoogle Scholar
[3]Belitskii, G.. The equivalence of families of local diffeomorphisms. Funct. Anal. Appl. 8 (1974), 338–339.CrossRefGoogle Scholar
[4]Belitskii, G.. Borel’s theorem in a Banach space. Math. Phys. Funct. Anal. 148 (1986), 137–139.Google Scholar
[5]Belitskii, G.. Normal Forms, Invariant and Local Mappings. Naukova Dumka, Kiev, 1979.Google Scholar
[6]Brjuno, A.. Analytic form of Differential Equations (Transactions of the Moscow Mathematical Society for the Year 1971, 25). 1971.Google Scholar
[7]Chen, K.. Equivalence and decomposition of vector fields about an elementary critical points. Amer. J. Math. 85 (1968), 693–722.CrossRefGoogle Scholar
[8]Hartman, P.. On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana 5(2) (1960), 220–241.Google Scholar
[9]Nitecki, Z.. Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms. MIT Press, Cambridge, MA, 1971.Google Scholar
[10]Rayskin, V.. Smooth dependence of linearization on perturbations. Mathematical Physics Preprint Archive, mp_arc 06–92 (2006).Google Scholar
[11]Rodrigues, H. and Solà-Morales, J.. Invertible contractions and asymptotically stable ODE’S that are not C 1-linearizable. J. Dynam. Differential Equations 18 (2006), 961–974.CrossRefGoogle Scholar
[12]Rodrigues, H. and Solà-Morales, J.. Linearization of class C 1 for contractions on Banach spaces. J. Differential Equations 201 (2004), 351–382.CrossRefGoogle Scholar
[13]Sternberg, S.. Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809–824.CrossRefGoogle Scholar