# 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 $t\not=x$)

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

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 QuasisymmetricMapping 2013-03-22 14:06:45 2013-03-22 14:06:45 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 30C65 msc 26A15 msc 26A12 QuasiconformalMapping BeurlingAhlforsQuasiconformalExtension $M$-condition quasisymmetric $M$-quasisymmetric