Decreasing dilatation can increase dimension (Q938808)

In the \(K\)-quasiconformal (qc) mapping theory, one of the principal objects of study is the class of \(K\)-quasispheres (i.e. images of the unit sphere \(S^{n-1} \subset {\mathbb R}^n\) under \(K\)-qc maps \(f: {\mathbb R}^n \to {\mathbb R}^n\)). For \(n=2\) these are called \(K\)-quasicircles. Very recently this topic was studied by many authors, e.g. by \textit{I. Prause} [Comput. Methods Funct. Theory 7, No. 2, 527--541 (2007; Zbl 1135.30309)]. It is a well-known fact that for each \(K>1\) there exist snowflake type \(K\)-quasicircles which are locally non-rectifiable. On the other hand if \(K\to 1\,,\) a \(K\)-quasicircle \(C\) becomes more flat in the sense that there exists a number \(r_0>0\) such that for every \(x\in C\) and every disk \(B^2(x,r), r\in (0, r_0),\) the set \(C \cap B^2(x,r)\) is contained in a rectangle centered at \(x\) with sidelengths \(2r\) and \(c_1 (K-1)r\) for some \(c_1>0\) (Prause). \textit{D. Meyer} constructed examples of snowflake type surfaces in \( {\mathbb R}^3 \,\) that can be quasiconformally mapped onto the unit ball under a quasiconformal mapping \( f:{\mathbb R}^3 \to {\mathbb R}^3 \,\) [Snowballs are Quasiballs, preprint, \url{arXiv:0810.2711}, to appear in Trans. Am. Math. Soc.]. The present author studies a problem of \textit{G. Cui} and \textit{M. Zinsmeister} [Ill. J. Math. 48, No. 4, 1223--1233 (2004; Zbl 1063.30015)] dealing with quasiconformal maps of the closed unit disk \( \overline{\mathbb D}\) with complex dilatation \(\mu \) such that \(f_{\mu}(\partial {\mathbb D})\) is a bilipschitz image of the circle. Is the same true for \(f_{t\mu}, 0<t<1?\) The author settles this question in the negative even in the case when \( f_{\mu} (\partial {\mathbb D})\) is a circle. A highly ingenious example is constructed such that \(\dim (f_{t \mu}(\partial {\mathbb D})) >1.\)