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Transformations preserving the Hausdorff-Besicovitch dimension |
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| Authors: | Sergio Albeverio,Grygoriy Torbin |
| Affiliation: | 1.Institut für Angewandte Mathematik,Universit?t Bonn,Bonn,Germany;2.SFB 611,Bonn,Germany;3.BiBoS,Bielefeld-Bonn,Germany;4.CERFIM,Locarno,Switzerland;5.Acc. Arch., USI,Locarno,Switzerland;6.IZKS,Bonn,Germany;7.National Pedagogical University,Kyiv,Ukraine;8.Institute for Mathematics of NASU,Kyiv,Ukraine |
| 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. |
| Keywords: | Hausdorff-Besicovitch dimension fractals transformations preserving the fractal dimension singularly continuous measures relative entropy of distributions |
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