# expansive

If $(X,d)$ is a metric space, a homeomorphism $f:X\to X$ is said to be expansive if there is a constant ${\epsilon}_{0}>0$, called the expansivity constant, such that for any two points of $X$, their $n$-th iterates are at least ${\epsilon}_{0}$ apart for some integer $n$; i.e. if for any pair of points $x\ne y$ in $X$ there is $n\in \mathbb{Z}$ such that $d({f}^{n}(x),{f}^{n}(y))\ge {\epsilon}_{0}$.

The space $X$ is often assumed to be compact^{}, since under that assumption^{} expansivity is a topological property; i.e. any map which is topologically conjugate to $f$ is expansive if $f$ is expansive (possibly with a different expansivity constant).

If $f:X\to X$ is a continuous map, we say that $X$ is positively expansive (or forward expansive) if there is ${\epsilon}_{0}$ such that, for any $x\ne y$ in $X$, there is $n\in \mathbb{N}$ such that $d({f}^{n}(x),{f}^{n}(y))\ge {\epsilon}_{0}$.

Remarks. The latter condition is much stronger than expansivity. In fact, one can prove that if $X$ is compact and $f$ is a positively expansive homeomorphism, then $X$ is finite (proof (http://planetmath.org/OnlyCompactMetricSpacesThatAdmitAPostivelyExpansiveHomeomorphismAreDiscreteSpaces)).

Title | expansive |
---|---|

Canonical name | Expansive |

Date of creation | 2013-03-22 13:47:48 |

Last modified on | 2013-03-22 13:47:48 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 14 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 37B99 |

Defines | expansivity |

Defines | positively expansive |

Defines | forward expansive |