Probability Theory and Related Fields

, Volume 97, Issue 4, pp 559–574

Self-similar random measures are locally scale invariant

  • N. Patzschke
  • M. Zähle
Article

Summary

In an earlier paper Patzschke and U. Zähle [11] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure Φ under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing Φ on the boundary of the open set with probability 1. The tangent measure isD-scale-invariant, whereD is the similarity dimension of Φ. Moreover, we approximate the tangential distribution by means of Φ and use this in order to prove that the Hausdorff dimension of the tangent measure equalsD. Since the former coincides with the Hausdorff dimension of Φ we obtain an earlier result of Mauldin and Williams [9] as a corollary.

Mathematics Subject Classification

60G57 60D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arbeiter, M.: Random recursive construction of self-similar fractal measures. The noncompact case. Probab. Theory Relat. Fields88, 497–520 (1991)Google Scholar
  2. 2.
    Bandt, C., Graf, S.: Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Am. Math. Soc.114, 995–1001 (1992)Google Scholar
  3. 3.
    Falconer, K.J.: Random fractals. Math. Proc. Camb. Philos. Soc.100, 559–582 (1986).Google Scholar
  4. 4.
    Feller, W.: An introduction to probability theory and its applications. II. New York: Wiley 1966Google Scholar
  5. 5.
    Graf, S.: Statistically self-similar fractals. Probab. Theory Relat. Fields74, 357–392 (1987)Google Scholar
  6. 6.
    Graf, S., Mauldin, R.D., Williams, S.C.: The exact Hausdorff dimension in random recursive constructions. Mem. Am. Math. Soc.71, no. 381 (1988)Google Scholar
  7. 7.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J.30, 713–747 (1981)Google Scholar
  8. 8.
    Kallenberg, O.: Random measures. Berlin: Akademie-Verlag & London: Academic Press 1983Google Scholar
  9. 9.
    Mauldin, R.D., Williams, S.C.: Random recursive constructions: Asymptotic geometric and topological properties. Trans. Am. Math. Soc.295, 325–346 (1986)Google Scholar
  10. 10.
    Moran, P.A.P.: Additive functions of intervals and Hausdorff measure. Proc. Camb. Philos. Soc.42, 15–23 (1946)Google Scholar
  11. 11.
    Patzschke, N., Zähle, U.: Self-similar random measures. IV. The recursive construction model of Falconer, Graf, and Mauldin and Williams. Math. Nachr.149, 285–302 (1990)Google Scholar
  12. 12.
    Zähle, M.: Ergodic theorems for general Palm measures. Math. Nachr.95, 93–106 (1980)Google Scholar
  13. 13.
    Zähle, U.: Self-similar random measures. I. Notion, carrying Hausdorff dimension and hyperbolic distribution. Probab. Theory Relat. Fields80, 79–100 (1988)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • N. Patzschke
    • 1
  • M. Zähle
    • 1
  1. 1.Mathematische FakultätFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations