Structural stability theorem

Let $M$ be a compact differentiable manifold and let $f:M\to M$ 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, $r\ge 2$ is still open.