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On the $ G$-isomorphism of probability and dimensional theories of representations of real numbers and fractal faithfulness of systems of coverings


Authors: I. I. Garko, R. O. Nikiforov and G. M. Torbin
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
Journal: Theor. Probability and Math. Statist. 94 (2017), 17-36
MSC (2010): Primary 11K55, 26A30, 28A80, 60G30
DOI: https://doi.org/10.1090/tpms/1006
Published electronically: August 25, 2017
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Abstract: A new method is developed to construct the probabilistic and dimensional theories for families of representations of real numbers based on studies of special mappings that preserve the Lebesgue measure and Hausdorff-Besicovitch dimension. These mappings are characterized by the property that a preimage and image have the same symbols for two representations of the same family (the set of points of discontinuity of such mappings can be everywhere dense). These mappings are said to be $ G$-mappings ($ G$-isomorphisms of representations). The probabilistic, metric, and dimensional theories of $ G$-isomorphic representations are identical. We establish a rather deep connection between the faithfulness of systems of coverings generated by different representations and the property of preservation of the Hausdorff-Besicovitch dimension of sets by the above-mentioned mappings. General sufficient conditions on faithfulness are found to evaluate the Hausdorff-Besicovitch dimension of families of cylinder sets generated by $ F$ and $ I$-$ F$-representations of real numbers.


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Additional Information

I. I. Garko
Affiliation: Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogov Street, 9, Kyiv 01130, Ukraine
Email: garko.iryna@gmail.com

R. O. Nikiforov
Affiliation: Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogov Street, 9, Kyiv 01130, Ukraine
Email: rnikiforov@gmail.com

G. M. Torbin
Affiliation: Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogov Street, 9, Kyiv 01130, Ukraine; Department of Fractal Analysis, Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 4, Kyiv 01130, Ukraine
Email: torbin7@gmail.com, torbin@npu.edu.ua

DOI: https://doi.org/10.1090/tpms/1006
Keywords: Fractals, DP-transformations, $G$-isomorphism of numeral systems, $F$-representations, $I$-$F$-representations, $Q_{\infty}$-representations, $\IQ$-representations, faithful covering systems, singularly continuous probability measures
Received by editor(s): May 5, 2016
Published electronically: August 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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