# $\mathrm{\Omega}$-stability theorem

Let $M$ be a differentiable manifold and let $f:M\to M$ be a ${\mathcal{C}}^{k}$ diffeomorphism. We say that $f$ is ${\mathcal{C}}^{k}$-$\mathrm{\Omega}$-stable, if there is a neighborhood^{} $\mathcal{U}$ of $f$ in the ${\mathcal{C}}^{k}$ topology of ${\mathrm{Diff}}^{k}(M)$ such that for any $g\in \mathcal{U}$, ${f|}_{\mathrm{\Omega}(f)}$ is topologically conjugate^{} to ${g|}_{\mathrm{\Omega}(g)}$.

$\mathrm{\Omega}$-stability theorem. If $f$ is Axiom A and satisfies the no-cycles condition, then $f$ is ${\mathcal{C}}^{1}$-$\mathrm{\Omega}$-stable.

*Remark.* The reciprocal of this theorem is also true (the difficult part is showing that $\mathrm{\Omega}$-stability implies Axiom A), but it is unknown whether ${\mathcal{C}}^{k}$-$\mathrm{\Omega}$-stability implies Axiom A when $k>1$. This is known as the ${\mathcal{C}}^{k}$ *$\mathrm{\Omega}$-stability conjecture*.

Title | $\mathrm{\Omega}$-stability theorem |
---|---|

Canonical name | OmegastabilityTheorem |

Date of creation | 2013-03-22 14:30:55 |

Last modified on | 2013-03-22 14:30:55 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 8 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 37C75 |

Synonym | omega-stability theorem |

Synonym | Smale’s $\mathrm{\Omega}$-stability theorem |

Defines | $\mathrm{\Omega}$-stable |

Defines | omega-stable |

Defines | $\mathrm{\Omega}$-stability |

Defines | omega-stability |