On the \(C^ 1\Omega\)-stability conjecture (Q1105249)

Let M be a compact \(C^{\infty}\) manifold without boundary. Denote by \(\Omega\) (f) the nonwandering set of a diffeomorphism f of M. A diffeomorphism is said to be \(C^ r\Omega\)-stable if there exists a neighbourhood U of f (in the C r topology) such that for each \(g\in U\) there exists a homeomorphism \(h: \Omega(f)\to \Omega(g)\) with \(g\circ h = h\circ f\) on \(\Omega(f)\). The main result of the present paper is: If a \(C^ 1\) diffeomorphism f is \(\Omega\)-stable then it satisfies axiom A. - As is explained in the papeer, this result constitutes the final step towards an affirmative answer of the \(C^ r\Omega\)-stability conjecture for \(r=1\). The \(C^ r\Omega\)-stability conjecture asserts: A \(C^ r\) diffeomorphism f is \(C^ r\Omega\)-stable if and only if f satisfies Axiom A, and there are no cycles on \(\Omega(f)\).