# 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 |