# Structural stability theorem

Let $M$ be a compact differentiable manifold and let $f:M\rightarrow M$ be a $C^{r}$ diffeomorphism.

Then, $f$ is structurally stable in the $C^{1}$ topology if and only if it is Axiom A and satisfies the strong transversality condition. The question of knowing if this is valid for the $C^{r}$ topology, $r\geq 2$ is still open.

Title Structural stability theorem StructuralStabilityTheorem 2014-03-19 22:14:32 2014-03-19 22:14:32 Filipe (28191) Filipe (28191) 5 Filipe (28191) Theorem stability conjecture $\Omega$-stability theorem