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.
@article{bwmeta1.element.doi-10_2478_s11533-008-0007-y,
author = {Sergio Albeverio and Mykola Pratsiovytyi and Grygoriy Torbin},
title = {Transformations preserving the Hausdorff-Besicovitch dimension},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {119-128},
zbl = {1134.28007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0007-y}
}
Sergio Albeverio; Mykola Pratsiovytyi; Grygoriy Torbin. Transformations preserving the Hausdorff-Besicovitch dimension. Open Mathematics, Tome 6 (2008) pp. 119-128. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0007-y/
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