Obstructions to finitely smooth linearizations
type: Item
instance of: scholarly article
title: Obstructions to finitely smooth linearizations
zbMATH Open document ID: 0674.58006
DOI: 10.1007/BF01077424
publication date: 1988-01-01T00:00:00Z
review text: The following assertion concerning a finitely-smooth linear approximation \(\Lambda\) of a diffeomorphism \(F\) in \(\mathbb R^ n\) are proved: Every \(C^ k\)-germ of a diffeomorphism \(F(x)=\Lambda x+o(\| x\|^ k)\) \((x\in\mathbb R^ n)\) is adjoint to a linear one in the class \(C^ k\) iff \((\max | \lambda |)^ k<\min | \lambda |\) \((\lambda\in\text{spec}\, \Lambda)\). The condition is equivalent to \[ | \lambda_ i| \neq | \lambda |^ a=| \lambda_ 1|^{a_ 1}... | \lambda_ n|^{a_ n} \] \((\lambda_ i\in\text{spec}\, \Lambda\), \(a\in\mathbb R^ n_+\), \(| a| =k)\). For every \(k<\infty\), there are hyperbolic \(C^ k\)-germs in the plane which are not adjoint in \(C^ k\) to any \(C^{k+1}\)-germ.
reviewed by: J.Durdil
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 4104022
zbMATH Keywords: finitely-smooth linear approximation
zbMATH Keywords: diffeomorphism
zbMATH Keywords: germ
author: G. Belitskii
MaRDI profile type: Publication
cites work: Equivalence and Decomposition of Vector Fields About an Elementary Critical Point
cites work: Q5592596
cites work: Local Contractions and a Theorem of Poincare
cites work: Invariant normal forms of formal series
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4104022