For special classes of functions, Devaney's three attributes of chaos are not independent. For example, |

Banks, et.al. showed that |

* if f:X → X is continuous and distances can be measured on X, |

* then sensitivity to initial conditions is a consequence of Devaney's other two conditions. |

Vellekoop and Berglund showed that |

* for continuous functions taking an interval to itself, |

* the mixing condition (also called transitivity) implies the other two of Devaney's requirements. |

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