# quasisymmetric mapping

A function $\mu $ of the real line to itself is *quasisymmetric* (or *$M$-quasisymmetric*) if it satisfies the following $M$-condition.

There exists an $M$, such that for all $x,t$ (where $t\ne x$)

$$\frac{1}{M}\le \frac{\mu (x+t)-\mu (x)}{\mu (x)-\mu (x-t)}\le M.$$ |

Geometrically this means that the ratio of the length of the intervals $\mu [(x-t,x)]$ and $\mu [(x,x+t)]$ is bounded. This implies among other things that the function is one-to-one and continuous^{}.

For example powers (as long as you make them one-to-one by for example using an odd power, or defining them as $-{|x|}^{p}$ for negative $x$ and ${|x|}^{p}$ for positive $x$ where $p>0$) are quasisymmetric. On the other hand functions like ${e}^{x}-{e}^{-x}$, while one-to-one, onto and continuous, are not quasisymmetric. It would seem like a very strict condition, however it has been shown that there in fact exist functions that are quasisymmetric, but are not even absolutely continuous^{}.

Quasisymmetric functions are an analogue of quasiconformal mappings.

Title | quasisymmetric mapping |
---|---|

Canonical name | QuasisymmetricMapping |

Date of creation | 2013-03-22 14:06:45 |

Last modified on | 2013-03-22 14:06:45 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30C65 |

Classification | msc 26A15 |

Classification | msc 26A12 |

Related topic | QuasiconformalMapping |

Related topic | BeurlingAhlforsQuasiconformalExtension |

Defines | $M$-condition |

Defines | quasisymmetric |

Defines | $M$-quasisymmetric |