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

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 GG-mappings (GG-isomorphisms of representations). The probabilistic, metric, and dimensional theories of GG-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 FF and II-FF-representations of real numbers.