Constructive Approximation

, Volume 5, Issue 1, pp 69–87 | Cite as

Approximation of measures by Markov processes and homogeneous affine iterated function systems

  • John H. Elton
  • Zheng Yan


It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system.

We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onR n can be approximated by invariant measures for finite iterated function systems of this class if\(\hat \mu (t)/\hat \mu (a^T t)\) is positive definite for some nonzero contractive linear mapa:R n R n . Moments and cumulants are also discussed.

AMS classification

60J05 60J15 60F05 42A38 42A82 58F11 41A99 

Key words and phrases

Invariant measures Markov processes Iterated function systems Approximation of measures Fourier transform Positive definite 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BDEG]
    M. F.Barnsley, S.Demko, J.Elton, J.Geronimo (1988):Invariant measures for Markov processes arising from function iteration with place-dependent probabilities. Ann. Inst. H. Poincaré,24.Google Scholar
  2. [BD]
    M. F. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A,399:243–275.Google Scholar
  3. [BE]
    M. F. Barnsley, J. Elton (1988):A new class of Markov processes for image encoding. Adv. in Appl. Probab.,20:14–32.Google Scholar
  4. [BS]
    M. F.Barnsley, A.Sloan (preprint):Image compression.Google Scholar
  5. [BA]
    M.Berger, Y.Amit (preprint):Products of random affine maps.Google Scholar
  6. [Bi]
    P. Billingsley (1986): Probability and Measure, 2nd ed. New York: Wiley.Google Scholar
  7. [Bo]
    R. P. Boas (1967):Lipschitz behavior and integrability of characteristic functions. Ann. of Math. Statist.,38:32–36.Google Scholar
  8. [Dg]
    J. Dugundji (1966): Topology. Boston: Allyn and Bacon.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • John H. Elton
    • 1
  • Zheng Yan
    • 1
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations