Skip to main content
SpringerLink
Log in

Constructing quasisymmetric functions via graph-directed iterated function systems

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beurling A. and Ahlfors L., “The boundary correspondence under quasiconformal mappings,” Acta Math., 96, No. 1–2, 125–142 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  2. Kelingos J. A., “Boundary correspondence under quasiconformal mappings,” Michigan Math. J., 13, No. 2, 235–249 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  3. Tukia P. and Väisälä J., “Quasisymmetric embeddings of metric spaces,” Ann. Acad. Sci. Fenn. Ser. A1 Math., 5, No. 1, 97–114 (1980).

    MATH  Google Scholar 

  4. Tukia P., “The space of quasisymmetric mappings,” Math. Scand., 40, No. 1, 127–142 (1977).

    MATH  MathSciNet  Google Scholar 

  5. Tukia P., “Hausdorff dimension and quasisymmetric mappings,” Math. Scand., 65, No. 1, 152–160 (1989).

    MATH  MathSciNet  Google Scholar 

  6. Mauldin R. D. and Williams S. C., “Hausdorff dimension in graph directed constructions,” Trans. Amer. Math. Soc., 309, No. 2, 811–829 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  7. Tetenov A. V., “Self-similar Jordan arcs and the graph directed systems of similarities,” Siberian Math. J., 47, No. 5, 940–949 (2006).

    Article  MathSciNet  Google Scholar 

  8. Aseev V. V. and Tetenov A. V., “Application of the graph directed systems to constructing quasisymmetric functions,” in: Proceedings of the International Conference “Current Problems of Mathematics and Mechanics” [in Russian], Trudy Mat. Tsentra im. N. I. Lobachevskogo. Vol. 25, Kazan Mat. Obshch., Kazan, 2004, pp. 31–32.

    Google Scholar 

  9. Edgar G., Measure, Topology, and Fractal Geometry, Springer-Verlag, New York, Berlin, and Heidelberg (1995).

    Google Scholar 

  10. Aseev V. V., Tetenov A. V., and Kravchenko A. S., “On selfsimilar Jordan curves on the plane,” Siberian Math. J., 44, No. 3, 379–386 (2003).

    Article  MathSciNet  Google Scholar 

  11. Hutchinson J., “Fractals and self-similarity,” Indiana Math. J., 30, No. 5, 713–747 (1981).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. V. Aseev

Authors
  1. V. V. Aseev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Aseev.

Additional information

Original Russian Text Copyright © 2009 Aseev V. V.

The author was supported by the Russian Foundation for Basic Research (Grants 09-01-90202-Monga and 09-01-98-001).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 6, pp. 1203–1215, November–December, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aseev, V.V. Constructing quasisymmetric functions via graph-directed iterated function systems. Sib Math J 50, 947–957 (2009). https://doi.org/10.1007/s11202-009-0105-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-009-0105-5

Keywords

Search

Navigation