# homoclinic class

Let $M$ be a compact^{} smooth manifold^{} and $f:M\to M$ a diffeomorphism. The *homoclinic class* of a hyperbolic periodic point^{} $p$ of $f$, denoted $H(p,f)$, is the closure^{} of the set of transverse intersections^{} between the stable and unstable manifolds all points in the orbit of $p$; i.e.

$$H(p,f)=\mathrm{cl}\left(\bigcup _{n\in \mathbb{N}}{W}^{s}(p)\u22d4\bigcup _{n\in \mathbb{Z}}{W}^{u}(p)\right).$$ |

Homoclinic classes are topologically transitive, and the number of homoclinic classes is at most countable^{}. Moreover, generically (in the ${\mathcal{C}}^{1}$ topology^{} of $\mathrm{Diff}(M)$), they are pairwise disjoint and maximally transitive^{}.

Title | homoclinic class |
---|---|

Canonical name | HomoclinicClass |

Date of creation | 2013-03-22 14:07:30 |

Last modified on | 2013-03-22 14:07:30 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 6 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37C29 |