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# 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.

Keywords:

structural stability, Axiom A, strong transversality

Related:

$\Omega$-stability theorem

Synonym:

stability conjecture

Major Section:

Reference

Type of Math Object:

Theorem

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