Quasiconformal dimensions of self-similar fractals

Jeremy T. Tyson, Jang Mei Wu

Research output: Contribution to journalArticlepeer-review

Abstract

The Sierpinski gasket and other self-similar fractal subsets of ℝd, d ≥ 2, can be mapped by quasiconformal self-maps of ℝd onto sets of Hausdorff dimension arbitrarily close to one. In ℝd we construct explicit mappings. In ℝd, d ≥ 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.

Original languageEnglish (US)
Pages (from-to)205-258
Number of pages54
JournalRevista Matematica Iberoamericana
Volume22
Issue number1
DOIs
StatePublished - 2006

Keywords

  • Conformal dimension
  • Hausdorff dimension
  • Iterated function system
  • Quasiconformal map
  • Sierpinski gasket

ASJC Scopus subject areas

  • Mathematics(all)

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