Abstract
Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R 1 resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.
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Albeverio, S., Pratsiovytyi, M. & Torbin, G. Transformations preserving the Hausdorff-Besicovitch dimension. centr.eur.j.math. 6, 119–128 (2008). https://doi.org/10.2478/s11533-008-0007-y
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DOI: https://doi.org/10.2478/s11533-008-0007-y