# hyperbolic fixed point

Let $M$ be a smooth manifold^{}. A fixed point^{} $x$ of a diffeomorphism
$f:M\to M$ is said to be a hyperbolic fixed point^{} if $Df(x)$ is a linear hyperbolic isomorphism. If $x$ is a periodic point of least period $n$, it is called a hyperbolic periodic point if it is a hyperbolic fixed point of ${f}^{n}$ (the $n$-th iterate of $f$).

If the dimension^{} of the stable manifold of a fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle.

Title | hyperbolic fixed point |

Canonical name | HyperbolicFixedPoint |

Date of creation | 2013-03-22 13:47:57 |

Last modified on | 2013-03-22 13:47:57 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37C25 |

Classification | msc 37D05 |

Related topic | StableManifold |

Related topic | HyperbolicSet |

Defines | hyperbolic periodic point |

Defines | source |

Defines | sink |

Defines | saddle |