# 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 ExampleOfAStrictlyIncreasingQuasisymmetricSingularFunction 2013-03-22 14:10:37 2013-03-22 14:10:37 jirka (4157) jirka (4157) 5 jirka (4157) Example msc 26A30