quasisymmetric mapping

A function μ 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 tx)


Geometrically this means that the ratio of the length of the intervals μ[(x-t,x)] and μ[(x,x+t)] is bounded. This implies among other things that the function is one-to-one and continuousMathworldPlanetmathPlanetmath.

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 ex-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 continuousMathworldPlanetmath.

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