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The \(C^{1+\alpha}\) hypothesis in Pesin theory. (English) Zbl 0542.58027
The stable manifold theory developed by Pesin and others contains a hypothesis that the given dynamics be of class \(C^{1+\alpha}\) for some \(\alpha>0\), i.e. the first derivatives must obey \(\alpha\)-Hölder conditions. This paper shows that \(C^ 1\) alone, i.e. \(\alpha =0\), is insufficient as follows. Theorem: There exists a \(C^ 1\) diffeomorphism of a 4-manifold having an asymptotically hyperbolic orbit O(p) such that \(W^ s(p)\) is not an injectively immersed manifold tangent to \(E^ s_ p\).
Reviewer: G.Ikegami

MSC:
37D99 Dynamical systems with hyperbolic behavior
58C25 Differentiable maps on manifolds
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