A proof of the \(C^ 1\) stability conjecture (Q1124188)

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.