Smale's Ω-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. Its non-wandering set has a spectral decomposition $\Omega=\bigsqcup_i\Omega_i$ into basic pieces. The no-cycle condition means that a cyclic chain of intersections $W^u(\Omega_i)\cap W^s(\Omega_{i+1})\ne\varnothing$ and $W^u(\Omega_n)\cap W^s(\Omega_1)\ne\varnothing$ can occur only when all the involved basic pieces coincide. Smale's Ω-stability conjecture. A $\mathcal C$-diffeomorphism is structurally stable if and only if it satisfies axiom A and the no-cycle condition. This is stated as a conjecture concerning the characterization of structural stability by axiom A and absence of cycles; the supplied text gives no resolution status.