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# Theory of Probability and Mathematical Statistics (Teoriya Imovirnostei ta Matematychna Statystyka)

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

### I. I. Garko, R. O. Nikiforov, G. M. Torbin

Abstract: We develop a new method for the construction of metric, probabilistic and dimensional theories for families of representations of real numbers via studies of spacial mappings, under which symbols of a given representation are mapped into the same symbols of other representation from the same family, and they preserve the Lebesgue measure and the Hausdorff-Besicovitch dimension (for such mappings the set of points of discontinuity can be everywhere dense). These mappings are said to be G-mappings (G-isomorphisms of representations). Probabilistic, metric and dimensional theories of G-isomorphic representations are identical. We show a rather deep connection between the faithfulness of systems of coverings, generated by different representations, and the preservation of the Hausdorff-Besicovitch dimension of sets by the above mentioned mappings. A special attention is paid to the development of probabilistic and dimensional theories of I-Q-representations of real numbers and to methods for pro ving of the Hausdorff faithfulness of coverings.

Keywords: fractal, DP-transformation, G-isomorphism of representations of real numbers, F-representation, I-F-representation, $Q_{\infty}$--representation, I-Q-representation, faithful system of coverings, singular continuous probability measure

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