# topologically transitive

A continuous surjection $f$ on a topological space^{} $X$ to itself
is *topologically transitive* if for every
pair of open sets $U$ and $V$ in $X$ there is an integer $n>0$
such that ${f}^{n}(U)\cap V\ne \mathrm{\varnothing}$, where ${f}^{n}$ denotes the $n$-th iterate of $f$.

If for every pair of open sets $U$ and $V$ there is an integer $N$ such that
${f}^{n}(U)\cap V\ne \mathrm{\varnothing}$ for each $n>N$, we say that $f$ is *topologically mixing*.

If $X$ is a compact^{} metric space, then $f$ is topologically transitive if and only if there exists a point $x\in X$ with a dense orbit, i.e. such that $\mathcal{O}(x,f)=\{{f}^{n}(x):n\in \mathbb{N}\}$ is dense in $X$.

Title | topologically transitive |
---|---|

Canonical name | TopologicallyTransitive |

Date of creation | 2013-03-22 13:41:05 |

Last modified on | 2013-03-22 13:41:05 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 5 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37B99 |

Classification | msc 54H20 |

Defines | topologically mixing |

Defines | topological mixing |