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Lowering the Assouad dimension by quasisymmetric mappings. (English) Zbl 0989.30017

Quasiconformal maps \(f\) of \(\mathbb{R}^n\) can distort Hausdorff dimension [F. W. Gehring, J. Väisälä J. Lond. Math. Soc., II. Ser. 6, 504-512 (1973; Zbl 0258.30020)], for more exact results in the plane see [K. Astala, Acta Math. 173, 37-60 (1994; Zbl 0815.30015)]. Similar question can be asked for quasisymmetric maps \(f:X\to Y\) between metric spaces \(X\) and \(Y\) and for other dimensions. The author considers the Assouad dimension, connected to the homogeneity of the space, see [J. Luukkainen, J. Korean Math. Soc. 35, No. 1, 23-76 (1998; Zbl 0893.54029)], and shows that spaces of Assouad dimension strictly less than one can be quasisymmetrically deformed onto spaces of arbitrary small Assouad dimension. The corresponding result for the Hausdorff dimension is not known. It is also shown that for each \(n\geq 1\) and a bounded set \(E\subset\mathbb{R}^n\) the global conformal Assouad dimension of \(E\) is either zero or \(\geq 1\) [P. Pansu, Ann. Acad. Sci. Fenn., Ser. A I 14, No. 2, 177-212 (1989; Zbl 0722.53028)]. The proofs and constructions make use of the connection of Assouad dimension to porosity.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
54E40 Special maps on metric spaces
54F45 Dimension theory in general topology
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