David maps and Hausdorff Dimension

Details David maps and Hausdorff Dimension David maps and Hausdorff Dimension

2002-12-07

arxiv.org/abs/math/0212106v1

Mathematics

Dynamical Systems

15 Pages, 5 figures

Scientific paper

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 {enumerate} [$\bullet$] Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David map $\phi:\CC \to \CC$ and a compact set $\Lambda$ such that $\Hdim \Lambda =\alpha$ and $\Hdim \phi(\Lambda)=\beta$. \vs [$\bullet$] There exists a David map $\phi:\CC \to \CC$ such that the Jordan curve $\Gamma=\phi (\Sen)$ satisfies $\Hdim \Gamma=2$.\vs {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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