Transformations preserving the Hausdorff-Besicovitch dimension

Sergio Albeverio; Mykola Pratsiovytyi; Grygoriy Torbin

Open Mathematics (2008)

  • Volume: 6, Issue: 1, page 119-128
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Sergio Albeverio, Mykola Pratsiovytyi, and Grygoriy Torbin. "Transformations preserving the Hausdorff-Besicovitch dimension." Open Mathematics 6.1 (2008): 119-128. <http://eudml.org/doc/269012>.

@article{SergioAlbeverio2008,
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.},
author = {Sergio Albeverio, Mykola Pratsiovytyi, Grygoriy Torbin},
journal = {Open Mathematics},
keywords = {Hausdorff-Besicovitch dimension; fractals; transformations preserving the fractal dimension; singularly continuous measures; relative entropy of distributions},
language = {eng},
number = {1},
pages = {119-128},
title = {Transformations preserving the Hausdorff-Besicovitch dimension},
url = {http://eudml.org/doc/269012},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Sergio Albeverio
AU - Mykola Pratsiovytyi
AU - Grygoriy Torbin
TI - Transformations preserving the Hausdorff-Besicovitch dimension
JO - Open Mathematics
PY - 2008
VL - 6
IS - 1
SP - 119
EP - 128
AB - 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.
LA - eng
KW - Hausdorff-Besicovitch dimension; fractals; transformations preserving the fractal dimension; singularly continuous measures; relative entropy of distributions
UR - http://eudml.org/doc/269012
ER -

References

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  1. [1] Albeverio S., Pratsiovytyi M., Torbin G., Fractal probability distributions and transformations preserving the Hausdorff-Besicovitch dimension, Ergodic Theory Dynam. Systems, 2004, 24, 1–16 http://dx.doi.org/10.1017/S0143385703000397 Zbl1115.37016
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  13. [13] Rogers C.A., Hausdorff measures, Cambridge University Press, Cambridge, 1998 Zbl0915.28002
  14. [14] Salem R., On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 1943, 53, 423–439 http://dx.doi.org/10.2307/1990210 Zbl0060.13709
  15. [15] Sauer T.D., Yorke J.A., Are the dimensions of a set and its image equal under typical smooth functions?, Ergodic Theory Dynam. Systems, 1997, 17, 941–956 http://dx.doi.org/10.1017/S0143385797086252 Zbl0884.28006
  16. [16] Turbin A.F., Pratsiovytyi M.V., Fractal sets functions and distributions, Naukova Dumka, Kiev, 1992 

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