# $\Omega$-stability theorem

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

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

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

Title $\Omega$-stability theorem OmegastabilityTheorem 2013-03-22 14:30:55 2013-03-22 14:30:55 Koro (127) Koro (127) 8 Koro (127) Theorem msc 37C75 omega-stability theorem Smale’s $\Omega$-stability theorem $\Omega$-stable omega-stable $\Omega$-stability omega-stability