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Normal hyperbolicity and linearisability. (English) Zbl 0619.58033
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 C. Pugh and 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.
Reviewer: A.Reinfelds (Riga)

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior
##### Keywords:
diffeomorphism; normal hyperbolicity
Full Text:
##### References:
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