# recurrent point

Let $X$ be a Hausdorff space and $f:X\to X$ a function. A point $x\in X$ is said to be *recurrent* (for $f$) if $x\in \omega (x)$, i.e. if $x$ belongs to its $\omega $-limit (http://planetmath.org/OmegaLimitSet3) set. This means that for each neighborhood^{} $U$ of $x$ there exists $n>0$ such that ${f}^{n}(x)\in U$.

The closure^{} of the set of recurrent points of $f$ is often denoted $R(f)$ and is called the *recurrent set* of $f$.

Every recurrent point is a nonwandering point, hence if $f$ is a homeomorphism^{} and $X$ is compact^{}, $R(f)$ is an invariant subset of $\mathrm{\Omega}(f)$, which may be a proper subset^{}.

Title | recurrent point |
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Canonical name | RecurrentPoint |

Date of creation | 2013-03-22 14:29:53 |

Last modified on | 2013-03-22 14:29:53 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 10 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37B20 |

Related topic | NonwanderingSet |

Defines | recurrent set |