Revista Matemática Iberoamericana

Quasiconformal dimensions of self-similar fractals

Jeremy T. Tyson and Jang-Mei Wu

Full-text: Open access


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

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]

quasiconformal map Hausdorff dimension conformal dimension Sierpinski gasket iterated function system


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

Export citation


  • Ahlfors, L. V.: Quasiconformal reflections. Acta Math. 109 (1963), 291-301.
  • Astala, K.: Area distortion of quasiconformal mappings. Acta Math. 173 (1994), no. 1, 37-60.
  • Balogh, Z. M.: Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group. J. Anal. Math. 83 (2001), 289-312.
  • Bishop, C. J.: Quasiconformal mappings which increase dimension. Ann. Acad. Sci. Fenn. Math. 24 (1999), 397-407.
  • Bishop, C. J. and Tyson, J. T.: Conformal dimension of the antenna set. Proc. Amer. Math. Soc. 129 (2001), 3631-3636.
  • Bishop, C. J. and Tyson, J. T.: Locally minimal sets for conformal dimension. Ann. Acad. Sci. Fenn. Math. 26 (2001), 361-373.
  • Bonk, M. and Kleiner, B.: Conformal dimension and Gromov hyperbolic groups with $2$-sphere boundary. Geom. Topol. 9 (2005), 219-246.
  • Bonk, M. and Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150 (2002), no. 1, 127-183.
  • Gehring, F. W. and Väisälä, J.: Hausdorff dimension and quasiconformal mappings. J. London Math. Soc. (2) 6 (1973), 504-512.
  • Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.
  • Hutchinson, J. E.: Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713-747.
  • Iwaniec, T. and Martin, G.: Geometric function theory and non-linear analysis. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2001.
  • Keith, S. and Laakso, T.: Conformal Assouad dimension and modulus. Geom. Funct. Anal. 14 (2004), no. 6, 1278-1321.
  • Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 335 (1993), no. 2, 721-755.
  • Kigami, J.: Analysis on fractals. Cambridge Tracts in Mathematics 143. Cambridge University Press, Cambridge, 2001.
  • Laakso, T.: personal communication.
  • Mañé, R. Sad, P. and Sullivan, D.: On the dynamics of rational maps. Ann. Sci. École Norm. Sup. (4) 16 (1983), 193-217.
  • MacManus, P.: Catching sets with quasicircles. Rev. Mat. Iberoamericana 15 (1999), no. 2, 267-277.
  • Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge, 1995.
  • Meyer, D.: personal communication.
  • Moran, P. A. P.: Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15-23.
  • Pansu, P.: Dimension conforme et sphère à l'infini des variétés à courbure négative. Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212.
  • Slodkowski, Z.: Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 347-355.
  • Tukia, P. and Väisälä, J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114.
  • Tukia, P. and Väisälä, J.: Extension of embeddings close to isometries or similarities. Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 153-175.
  • Tyson, J. T.: Sets of minimal Hausdorff dimension for quasiconformal maps. Proc. Amer. Math. Soc. 128 (2000), no. 11, 3361-3367.
  • Tyson, J. T.: Lowering the Assouad dimension by quasisymmetric mappings. Illinois J. Math. 45 (2001), 641-656.
  • Väisälä, J.: Bi-Lipschitz and quasisymmetric extension properties. Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 2, 239-274.
  • Väisälä, J.: Quasisymmetry and unions. Manuscripta Math. 68 (1990), 101-111.