Normal hyperbolicity and linearisability (Q1089659)

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.