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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: vrayskin@gmail.com)
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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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