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The probability distribution and Hausdorff dimension of self-affine functions
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  • Published: September 1990

The probability distribution and Hausdorff dimension of self-affine functions

  • M. Urbański1 nAff2 

Probability Theory and Related Fields volume 84, pages 377–391 (1990)Cite this article

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  • 11 Citations

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Summary

A simple natural measure is found with respect to which the probability distribution of a continuous self-affine functionf in the sense of Kôno is absolutely continuous. As an immediate corollary we obtain the result of Kôno that provides a necessary and sufficient condition for this distribution to be absolutely continuous with respect to Lebesgue measure. For the class of continuous self-affine functions one proves the conjecture of T. Bedford which says in this context that the Hausdorff dimension of the graph off is equal to its box dimension if and only if the probability distribution off is absolutely continuous with respect to Lebesgue measure.

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References

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Author information

Author notes
  1. M. Urbański

    Present address: Institut für Mathematische Stochastik, Georg-August-Universität, Lotzestrasse 13, D-3400, Göttingen, Federal Republic of Germany

Authors and Affiliations

  1. Instytut Matematyki, Uniwersytet M. Kopernika, ul. Chopina 12/18, 87-100, Toruń, Poland

    M. Urbański

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  1. M. Urbański
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Urbański, M. The probability distribution and Hausdorff dimension of self-affine functions. Probab. Th. Rel. Fields 84, 377–391 (1990). https://doi.org/10.1007/BF01197891

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  • Received: 21 June 1988

  • Revised: 08 March 1989

  • Issue Date: September 1990

  • DOI: https://doi.org/10.1007/BF01197891

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Keywords

  • Probability Distribution
  • Stochastic Process
  • Probability Theory
  • Lebesgue Measure
  • Mathematical Biology
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