Palis–Smale structural stability conjecture for diffeomorphisms

Let $1\le r\le\infty$ or let $\mathcal C=\mathcal H$, and let $\mathcal C$ be the corresponding category of maps. A $\mathcal C$-diffeomorphism is structurally stable if every $\mathcal C$-perturbation is topologically conjugate to it. An axiom A diffeomorphism satisfies the strong transversality condition when its stable and unstable manifolds intersect transversally. Palis–Smale structural stability conjecture. A $\mathcal C$-diffeomorphism is structurally stable if and only if it satisfies axiom A and the strong transversality condition. This is presented as an outstanding conjecture about structural stability; the supplied text gives no resolution status.