Nikiforov R. Multifractal analysis of singular probability measures

The thesis is devoted to the investigation of fractal properties of singularly continuous probability distributions and to applications of the results to the number theory, theory of dynamical systems, fractal geometry. The external multifractal analysis of probability measures and ergodic theory are theoretical basis for these investigations. We establish several new probabilistic, fractal and number theoretical phenomena connected with the Q-expansion. We prove sufficient conditions for the non-faithfulness of the family of Q-cylinders. On the other hand, sufficient conditions for the faithfulness of such covering systems are also found and ergodic properties of the Q-expansion are studied. We study properties of probability measures generated by random Q-expansions. An explicit formulae for the determination of the Hausdorff dimension of the corresponding probability measures is proven.