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.

Title	Structural stability theorem
Canonical name	StructuralStabilityTheorem
Date of creation	2014-03-19 22:14:32
Last modified on	2014-03-19 22:14:32
Owner	Filipe (28191)
Last modified by	Filipe (28191)
Numerical id	5
Author	Filipe (28191)
Entry type	Theorem
Synonym	stability conjecture
Related topic	$\mathrm{\Omega }$-stability theorem