example of a strictly increasing quasisymmetric singular function

An example of a strictly increasing quasisymmetric function that also a purely singular function can be defined as:

f(x)=\lim_{k\rightarrow\infty}\int_{0}^{x}\prod_{i=1}^{k}(1+\lambda\cos n_{i}s% )ds,

where $0<\lambda<1$ and carefully picked $n_{i}$ . We can pick the $n_{i}$ such that $n_{i+1}$ is strictly greater then $\sum_{j=1}^{i}n_{j}$ . However if we pick the $\lambda$ and $n_{i}$ more carefully, we can construct functions with the quasisymmetricity constant as close to 1 as we want. That is, we can construct functions such that

\frac{1}{M}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)}\leq M

for all $x$ and $t$ where $M$ is as close to 1 as we want. If $M=1$ note that the function must be a straight line.

It is also possible from this to construct a quasiconformal mapping of the upper half plane to itself by extending this function to the whole real line and then using the Beurling-Ahlfors quasiconformal extension. Then we’d have a quasiconformal mapping such that its boundary correspondence would be a purely singular function.

For more detailed explanation, and proof (it is too long to reproduce here) see bibliography.

Bibliography

•

A. Beurling, L. V. Ahlfors. . Acta Math., 96:125-142, 1956.
•

J. Lebl. . . Also available at http://www.jirka.org/thesis.pdfhttp://www.jirka.org/thesis.pdf

Title	example of a strictly increasing quasisymmetric singular function
Canonical name	ExampleOfAStrictlyIncreasingQuasisymmetricSingularFunction
Date of creation	2013-03-22 14:10:37
Last modified on	2013-03-22 14:10:37
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Example
Classification	msc 26A30