On conformal welding and quasicircles (Q802787)

The paper deals with the question how a certain geometric property of a quasicircle C is reflected by the conformal welding \(\phi: \pi\to \pi\), \(\phi =g^{-1}\circ f\), where f and g are conformal maps of the unit disk to the two components of \({\mathbb{C}}\setminus C\). This condition is shared by self-similar curves like the snowflake or (certain) Julia sets. The main result is that, assuming the geometric condition, the welding is ``strongly singular'' in the sense that there is a set \(E\subset \pi\) of Hausdorff-dimension \(<1\) that is mapped onto a set with \(\dim (\pi \setminus \phi (E))<1\). Examples of strongly singular weldings were first constructed by P. Tukia. The above result is applied to give a partial answer to a problem raised by Tukia concerning isomorphisms between Fuchsian groups.