## 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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