example of a strictly increasing quasisymmetric singular function
where and carefully picked . We can pick the such that is strictly greater then . However if we pick the and 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
for all and where is as close to 1 as we want. If 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.
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|
|Date of creation||2013-03-22 14:10:37|
|Last modified on||2013-03-22 14:10:37|
|Last modified by||jirka (4157)|