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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
##### Keywords:
stable manifold; diffeomorphism
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##### References:
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