JournalsrmiVol. 22, No. 1pp. 205–258

Quasiconformal dimensions of self-similar fractals

  • Jeremy T. Tyson

    University of Illinois, Urbana, USA
  • Jang-Mei Wu

    University of Illinois at Urbana-Champaign, USA
Quasiconformal dimensions of self-similar fractals cover
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Abstract

The Sierpinski gasket and other self-similar fractal subsets of Rd\mathbb R^d, d2d\ge 2, can be mapped by quasiconformal self-maps of Rd\mathbb R^d onto sets of Hausdorff dimension arbitrarily close to one. In R2\mathbb R^2 we construct explicit mappings. In Rd\mathbb R^d, d3d\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.

Cite this article

Jeremy T. Tyson, Jang-Mei Wu, Quasiconformal dimensions of self-similar fractals. Rev. Mat. Iberoam. 22 (2006), no. 1, pp. 205–258

DOI 10.4171/RMI/454