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:

$$f(x)=\underset{k\to \mathrm{\infty }}{lim}{\int }_0^x\prod _{i=1}^k(1+\lambda \cos n_{i}s)ds,$$

where and carefully picked $n_i$.We can pick the $n_i$ such that $n_{i+1}$ is strictlygreater then ${\sum }_{j=1}^i{n}_j$. However if we pick the $\lambda$and $n_i$ more carefully, we can construct functions with the quasisymmetricityconstant as close to 1 as we want. That is, we can construct functions such that

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

for all $x$ and $t$ where $M$ is as close to 1 as we want. If $M=1$ note thatthe 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, andproof (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 athttp://www.jirka.org/thesis.pdfhttp://www.jirka.org/thesis.pdf