Institute for Mathematical Sciences

Preprint ims02-05

S. Zakeri
David maps and Hausdorff Dimension

Abstract: David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs \begin{enumerate} \item[$\bullet$] Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David map $\varphi:\CC \to \CC$ and a compact set $\Lambda$ such that $\Hdim \Lambda =\alpha$ and $\Hdim \varphi(\Lambda)=\beta$. \vs \item[$\bullet$] There exists a David map $\varphi:\CC \to \CC$ such that the Jordan curve $\Gamma=\varphi (\Sen)$ satisfies $\Hdim \Gamma=2$.\vs \end{enumerate} One should contrast the first statement with the fact that quasiconformal maps preserve sets of Hausdorff dimension $0$ and $2$. The second statement provides an example of a Jordan curve with Hausdorff dimension $2$ which is (quasi)conformally removable.
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