Feshchenko O. Application of f-encoding of real numbers to investigation of fractal properties of probability measures

In the thesis we study structure, topological, metric and fractal properties of random variables (r.v.) such that symbols of certain their representation (encoding) either are independent or form the Markov chain. The G2-representation of real numbers is introduced. It is offer some new classes of fractal objects with simple formal analytic definition. For proposed encoding system we study ''normal'' properties of numbers in terms of of frequencies of symbols of G2-representation. For r.v. with independent G2-symbols we study the Lebesgue structure of distribution (i.e., content of discrete, absolutely continuous and singular component) as well as the structure of singular distribution (i.e., content of the Salem-, Cantor- and quasi-Cantor-type component). We study topological, metric and fractal properties of spectra of r.v. from this class. We also solve similar problems for one class of Bernoulli convolutions namely random incomplete sums of the convergent positive series with independent addenda.