David Maps and Hausdorff Dimension
نویسنده
چکیده
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: – Given α and β in [0, 2] , there exists a David map φ: C → C and a compact set Λ such that dimH Λ = α and dimH φ(Λ) = β . – There exists a David map φ: C → C such that the Jordan curve Γ = φ(S) satisfies dimH Γ = 2. 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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