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May, 2006 Quasiconformal dimensions of self-similar fractals
Jeremy T. Tyson , Jang-Mei Wu
Rev. Mat. Iberoamericana 22(1): 205-258 (May, 2006).

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.

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Jeremy T. Tyson . Jang-Mei Wu . "Quasiconformal dimensions of self-similar fractals." Rev. Mat. Iberoamericana 22 (1) 205 - 258, May, 2006.

Information

Published: May, 2006
First available in Project Euclid: 24 May 2006

zbMATH: 1108.30015
MathSciNet: MR2268118

Subjects:
Primary: 28A80 , 30C65 , 34C35
Secondary: 51M20

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

Rights: Copyright © 2006 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.22 • No. 1 • May, 2006
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