$\mathrm{\Omega }$-stability theorem

Let $M$ be a differentiable manifold and let $f:M\to M$ be a ${𝒞}^k$ diffeomorphism. We say that $f$ is ${𝒞}^k$-$\mathrm{\Omega }$-stable, if there is a neighborhood $𝒰$ of $f$ in the ${𝒞}^k$ topology of ${\mathrm{Diff}}^k(M)$ such that for any $g\in 𝒰$, ${f|}_{\mathrm{\Omega }(f)}$ is topologically conjugate to ${g|}_{\mathrm{\Omega }(g)}$.

$\mathrm{\Omega }$-stability theorem. If $f$ is Axiom A and satisfies the no-cycles condition, then $f$ is ${𝒞}^1$-$\mathrm{\Omega }$-stable.

Remark. The reciprocal of this theorem is also true (the difficult part is showing that $\mathrm{\Omega }$-stability implies Axiom A), but it is unknown whether ${𝒞}^k$-$\mathrm{\Omega }$-stability implies Axiom A when $k>1$. This is known as the ${𝒞}^k$ $\mathrm{\Omega }$-stability conjecture.