# periodic point

Let $f:X\to X$ be a function^{} and ${f}^{n}$ its $n$-th iteration. A point $x$ is called a periodic point^{} of period $n$ of $f$ if it is a fixed point of ${f}^{n}$. The least $n$ for which $x$ is a fixed point of ${f}^{n}$ is called prime period or least period.

If $f$ is a function $\mathbb{R}$ to $\mathbb{R}$ or $\u2102$ to $\u2102$ then a periodic point $x$ of prime period $n$ is called hyperbolic if $|{({f}^{n})}^{\prime}(x)|\ne 1$, attractive if $$ and repelling if $|{({f}^{n})}^{\prime}(x)|>1$.

Title | periodic point |
---|---|

Canonical name | PeriodicPoint |

Date of creation | 2013-03-22 12:43:38 |

Last modified on | 2013-03-22 12:43:38 |

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 14 |

Author | mathwizard (128) |

Entry type | Definition |

Classification | msc 26A18 |

Defines | hyperbolic periodic point |

Defines | attractive periodic point |

Defines | repelling periodic point |

Defines | least period |

Defines | prime period |