Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-18T15:49:53.033Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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