Revista Matemática Iberoamericana

Quasiconformal dimensions of self-similar fractals

Jeremy T. Tyson and Jang-Mei Wu

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Abstract

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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 1 (2006), 205-258.

Dates
First available in Project Euclid: 24 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1148492181

Mathematical Reviews number (MathSciNet)
MR2268118

Zentralblatt MATH identifier
1108.30015

Subjects
Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations 28A80: Fractals [See also 37Fxx] 34C35
Secondary: 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

Keywords
quasiconformal map Hausdorff dimension conformal dimension Sierpinski gasket iterated function system

Citation

Tyson , Jeremy T.; Wu , Jang-Mei. Quasiconformal dimensions of self-similar fractals. Rev. Mat. Iberoamericana 22 (2006), no. 1, 205--258. https://projecteuclid.org/euclid.rmi/1148492181


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