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Quasiconformal dimensions of selfsimilar fractals. (English) Zbl 1108.30015

Dilation independent bounds for Hausdorff dimension distortion under quasiconformal mappings are studied. C. Bishop [Ann. Acad. Sci. Fenn., Math. 24, No. 2, 397–407 (1999; Zbl 0945.30020)] showed that for sets of positive dimension and less than the dimension \(d\) of the target space there is never an obstruction to raising dimension. In the other direction for each \(\alpha \in [1, d]\) there exists a compact set \(E \subset \mathbb R^d\) for which \(\dim f(E) \geq \dim E = \alpha\) for every quasiconformal mapping \(f : \mathbb R^d \rightarrow \mathbb R^d\), see [C. J. Bishop and J. T. Tyson, Ann. Acad. Sci. Fenn., Math. 26, No. 2, 361–373 (2001; Zbl 1013.30015)].
For a fixed set \(E \subset \mathbb R^d\) the quasiconformal dimension \(\dim_{QC} E\) of \(E\) is the infimum of the Hausdorff dimensions of all images of \(E\) under quasiconformal self maps of \(\mathbb R^d\). The authors show that \(\dim_{QC} S = 1\) where \(S\) is the classical Sierpiński gasket in \(\mathbb R^d\). Related results for the conformal dimension have been obtained earlier. Iterated function systems are used to describe the self similar fractals and the constructions of the required quasiconformal mappings are based on extension properties of quasisymmetric mappings.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
28A80 Fractals
51M20 Polyhedra and polytopes; regular figures, division of spaces
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