# 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

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topSergio 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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