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


37C75 Stability theory for smooth dynamical systems
37C20 Generic properties, structural stability of dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Full Text: DOI Numdam


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