Revista Matemática Iberoamericana
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Quasiconformal dimensions of self-similar fractals
Jeremy T. Tyson (1) and Jang-Mei Wu (2)(1) Department of Mathematics, University of Illinois, 1409 W Green Street, IL 61801, URBANA, UNITED STATES
(2) Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green Street, IL 61801-2975, URBANA, UNITED STATES
The Sierpinski gasket and other self-similar fractal subsets of $\mathbb R^d$, $d\ge 2$, can be mapped by quasiconformal self-maps of $\mathbb R^d$ onto sets of Hausdorff dimension arbitrarily close to one. In $\mathbb R^2$ we construct explicit mappings. In $\mathbb R^d$, $d\ge 3$, the results follow from general theorems on the equivalence of invariant sets for iterated function systems under quasisymmetric maps and global quasiconformal maps. More specifically, we present geometric conditions ensuring that (i) isomorphic systems have quasisymmetrically equivalent invariant sets, and (ii) one-parameter isotopies of systems have invariant sets which are equivalent under global quasiconformal maps.
Keywords: Quasiconformal map, Hausdorff dimension, conformal dimension, Sierpinski gasket, iterated function system