Transformations preserving the Hausdorff-Besicovitch dimension

• Autorzy: Sergio Albeverio , Mykola Pratsiovytyi , Grygoriy Torbin
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Hausdorff-Besicovitch dimension; fractals; transformations preserving the fractal dimension; singularly continuous measures; relative entropy of distributions

Kategorie tematyczne

• 60G30: Continuity and singularity of induced measures
• 28A80: Fractals
• 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2008
• Tom: 6
• Numer: 1
• Strony: 119-128

Daty

wydano

2008-03-01

online

2008-02-26

Twórcy

autor

Sergio Albeverio

autor

Mykola Pratsiovytyi

autor

Grygoriy Torbin

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-008-0007-y