Sergio Albeverio, Vyacheslav Koval, Mykola Pratsiovytyi, G. Torbin
September 11, 2008

### Abstract

A multidimensional classification of singularly continuous (w.r.t. the Lebesgue measure) probability measures in R 2 is introduced and a theorem on canonical representation of such measures is proven. A class of random elements on the unit square which is defined by a system of partitions generated by the Q *-representation of real numbers is introduced and studied in details. Conditions for the discreteness, absolute continuity resp. singular continuity (w.r.t. Lebesgue measure) of the corresponding probability measures are found. Metric, topological and fractal properties of the spectra of the corresponding probability distributions are investigated. A class of transformations preserving the Hausdorff-Besicovitch dimension of any subset of the unit square (DP-transformations) is also studied.