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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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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DOI: https://doi.org/10.1007/BF01197891
Keywords
- Probability Distribution
- Stochastic Process
- Probability Theory
- Lebesgue Measure
- Mathematical Biology