Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Random recursive construction of self-similar fractal measures. The noncompact case
Download PDF
Download PDF
  • Published: December 1991

Random recursive construction of self-similar fractal measures. The noncompact case

  • Matthias Arbeiter1 

Probability Theory and Related Fields volume 88, pages 497–520 (1991)Cite this article

  • 127 Accesses

  • 28 Citations

  • Metrics details

Summary

The self-similarity of sets (measures) is often defined in a constructive way. In the present paper it will be shown that the random recursive construction model of Falconer, Graf and Mauldin/Williams for (statistically) self-similar sets may be generalized to the noncompact case. We define a sequence of random finite measures, which converges almost surely to a self-similar random limit measure. Under certain conditions on the generating Lipschitz maps we determine the carrying dimension of the limit measure.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • [F1] Falconer, K.J.: The geometry of fractal sets. Cambridge: Cambridge University Press 1985

    Google Scholar 

  • [F2] Falconer, K.J.: Random fractals. Math. Proc. Camb. Philos. Soc.100, 559–582 (1986)

    Google Scholar 

  • [Fe] Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969

    Google Scholar 

  • [GH] Geronimo, J.S., Hardin, D.P.: An exact formula for the measure dimensions associated with a class of piecewice linear maps. Contructive Approximation5, 89–98 (1989)

    Google Scholar 

  • [G] Graf, S.: Statistically self-similar fractals. Probab. Th. Rel. Fields74, 357–392 (1987)

    Google Scholar 

  • [GMW] Graf, S., Mauldin, R.D., Williams, S.C.: The exact Hausdorff dimension in random recursive constructions. Mem. Am. Math. Soc.381, 121 (1988)

    Google Scholar 

  • [H] Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J.30, 713–747 (1981)

    Google Scholar 

  • [K] Kallenberg, O.: Random measures. Berlin: Akademie 1983

    Google Scholar 

  • [M] Mandelbrot, B.B.: The fractal geometry of Nature. New York: Birkhäuser 1983/Berlin-Basel: Birkhäuser 1987

    Google Scholar 

  • [MW] Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Am. Math. Soc.295, 325–346 (1986)

    Google Scholar 

  • [PZ] Patzschke, N., Zähle, U.: Self-similar random measures. IV — The recursive construction model of Falcomer, Graf and Mauldin and Williams. Math. Nachr.149, 285–302 (1990)

    Google Scholar 

  • [R] Révész, P.: Die Gesetze der großen Zahlen. Akademiai Kiado Budapest/Basel: Birkhäuser 1968

    Google Scholar 

  • [Z1] Zähle, U.: The fractal character of localizable measure-valued processes. III — Fractal carrying sets of branching diffusions. Math. Nachr.138, 293–311 (1988)

    Google Scholar 

  • [Z2] Zähle, U.: Self-similar random measures. I — Notion, carrying Hausdorff dimension, and hyperbolic distribution. Probab. Th. Rel. Fields80, 79–100 (1988)

    Google Scholar 

  • [Z3] Zähle, U.: Self-similar random measures. II — Generalization to self-affine measures. Math. Nachr.146, 85–98 (1990)

    Google Scholar 

  • [Z4] Zähle, U.: Self-similar random measures. III — Self-similar random processes. Math. Nachr. (to appear)

Download references

Author information

Authors and Affiliations

  1. Sektion Mathematik, Friedrich-Schiller-Universität, O-6900, Jena, Germany

    Matthias Arbeiter

Authors
  1. Matthias Arbeiter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Arbeiter, M. Random recursive construction of self-similar fractal measures. The noncompact case. Probab. Th. Rel. Fields 88, 497–520 (1991). https://doi.org/10.1007/BF01192554

Download citation

  • Received: 02 May 1990

  • Revised: 02 December 1990

  • Issue Date: December 1991

  • DOI: https://doi.org/10.1007/BF01192554

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Construction Model
  • Limit Measure
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature