Revista Matemática Iberoamericana
- Rev. Mat. Iberoamericana
- Volume 22, Number 1 (2006), 205-258.
Quasiconformal dimensions of self-similar fractals
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.
Rev. Mat. Iberoamericana, Volume 22, Number 1 (2006), 205-258.
First available in Project Euclid: 24 May 2006
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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