Ratushniak S. Fractal functions and distributions of their values

The thesis is devoted to two classes of fractal functions defined on the unit segment in terms of one two-symbol generalization of the binary representation of numbers and its non-self-similar polybasic generalization. A function is a fractal function if the set of non-constancy points, the set of features (in particular, features of the differential nature), the level sets or the graph are fractal in a broad or narrow sense. Nowhere monotonic, nowhere differentiable and singular functions are examples of fractal functions. To define and study them a various systems of encoding of numbers are widely used. In this aspect, a non-trivial task is the analytical definition of a fractal function whose argument and value are given in terms of the same two-symbol representation. The main task of this work is an evaluation of the potential of two-symbol system of encoding of real numbers as an instrument for definition and studying functions with fractal properties, namely: using the functional dependence of two variables between pairs of digits of argument and digits of value of function to give a well-defined classes of functions with locally complicated structure. For two classes of functions related to the two-base encoding of numbers of a unit segment by means of a two-symbol alphabet, we study structural, variational, self-similar and fractal properties, describe the topological and metric properties of essential sets (graphs, level sets, sets of features of different kinds, etc.), and also study the Lebesgue structure of distributions of their values with uniform distribution of the argument. For constructively described classes of functions, fractal, differential, structural and variational properties are studied. The methodology of research of distributions of values of functions is shown on concrete examples. Two problems of probabilistic theory for encoding of numbers are solved. These results generalize the study of distributions of random variables given by their two-symbol representation.