Normal hyperbolicity and linearisability
type: Item
instance of: scholarly article
title: Normal hyperbolicity and linearisability
zbMATH Open document ID: 0619.58033
DOI: 10.1007/BF01389422
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143425
review text: It is well known that one can linearise a diffeomorphism near a compact invariant submanifold in the presence of 1-normal hyperbolicity. The author gives a counterexample to a statement suggested by \textit{C. Pugh} and \textit{M. Shub} [ibid. 10, 187-198 (1970; Zbl 0206.258)] that one can weaken this normal hyperbolicity assumption. Let \(M=S^ 3\) and consider \(V=S^ 2\) as a submanifold of M. The author proves the existence of \(C^{\infty}\) diffeomorphism \(f: M\to M\), which leaves V invariant and which is 0-normally hyperbolic along V such that (i) f is not conjugate to N(f) near V, (ii) if \(g: M\to M\) is a diffeomorphism with \(N(g)=N(f)\) along V and g is sufficiently \(C^ 2\) close to f, then g is not conjugate to N(g) near V.
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 4005203
zbMATH Keywords: diffeomorphism
zbMATH Keywords: normal hyperbolicity
author: Sebastian van Strien
reviewed by: Andrejs Reinfelds
MaRDI profile type: Publication
cites work: Invariant manifolds
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cites work: Q4179563
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description: scientific article; zbMATH DE number 4005203