Svynchuk O. Singular non-monotonic functions of Cantor type and their fractal properties

We consider the so-called Q*_s-representation of numbers xє [0,1]. It is an encoding of a number by means of finite alphabet A={0,\..., s-1} and generalizes s-adic and Q_s-representation of real numbers. The thesis is devoted to continuous on closed interval non-monotonic singular functions of Cantor type defined in terms of a given Q*_s-representation of numbers. We study their local and global properties: structural, variational, differential, integral, self-similar, and fractal. Level sets of function as well as topological and metric properties of images of Cantor type sets are examined in detail. In this work, we study the distribution of random variable Y=f(X), where f is a non-monotonic singular function of Cantor type and X is a random variable such that its distribution induced by distributions of digits of its Q*_5-representation that are independent random variables. Problems about Lebesgue structure of distribution (content of discrete, absolutely continuous and singular components) for random variable Y are considered. Generally speaking, distribution of random variable Y is a nontrivial mixture of discrete and continuous components (discrete and singular or discrete and absolutely continuous distributions). A criterion of pure discreteness as well as of pure continuity for this random variable is found. For some cases, the problem about Lebesgue structure of distribution is solved exhaustively.