# nonwandering set

Let $X$ be a metric space, and $f:X\to X$ a continuous^{} surjection.
An element $x$ of $X$ is a *wandering point* if there is a neighborhood^{} $U$ of $x$ and an integer $N$ such that, for all $n\ge N$, ${f}^{n}(U)\cap U=\mathrm{\varnothing}$. If $x$ is not wandering, we call it a *nonwandering point*. Equivalently, $x$ is a nonwandering point if for every neighborhood $U$
of $x$ there is $n\ge 1$ such that ${f}^{n}(U)\cap U$ is nonempty. The set of all nonwandering points is called the *nonwandering set* of $f$, and is denoted by $\mathrm{\Omega}(f)$.

If $X$ is compact^{}, then $\mathrm{\Omega}(f)$ is compact, nonempty, and forward invariant; if, additionally, $f$ is an homeomorphism^{}, then $\mathrm{\Omega}(f)$ is invariant.

Title | nonwandering set |
---|---|

Canonical name | NonwanderingSet |

Date of creation | 2013-03-22 13:39:31 |

Last modified on | 2013-03-22 13:39:31 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 4 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37B20 |

Related topic | OmegaLimitSet3 |

Related topic | RecurrentPoint |

Defines | wandering point |

Defines | nonwandering point |