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A proof of the $$C^ 1$$ stability conjecture. (English) Zbl 0678.58022
The fact that strong transversality plus Axiom A is a necessary and sufficient condition for a $$C^ r$$ diffeomorphism on a closed manifold to be $$C^ r$$ structurally stable is known as the Palais-Smale conjecture.
Sufficiency was proved in the 70’s by J. Robin and C. Robinson and necessity was reduced to proving that $$C^ r$$ structural stability implies Axiom A. This problem became known as the stability conjecture. The main result of this paper is the following:
Theorem: Every $$C^ 1$$ structurally stable diffeomorphism of a closed manifold satisfies Axiom A.
Reviewer: M.Teixeira

##### MSC:
 37C75 Stability theory for smooth dynamical systems 37C20 Generic properties, structural stability of dynamical systems 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
##### Keywords:
diffeomorphism; structural stability
Full Text:
##### References:
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