Klymchuk S. Fractal properties of asymptotic mean of digits function in s-adic representation of real number

The thesis is devoted to study function r of asymptotic mean of digits in s-adic representation of real number. We investigate its properties. In particular we study topological, metric and fractal properties of its level sets. In the thesis we analyze reversible connection between concepts of asymptotic mean of digits and frequency of digits in s-adic representation of real number (in the contexts of existence and relation). For each y in [0;s-1] we specify an algorithm of construction x in [0;1] such that y=r(x). We describe local and global properties of the function r. In the manuscript we consider transformations of [0;1) and functions preserving asymptotic mean of digits. We define its relationship with functions preserving the digit frequency and Hausdorff-Besicovitch fractal dimension. We construct examples of functions preserving asymptotic mean of digits and functions do not preserving the frequency of digits; also we construct functions preserving asymptotic mean of digits if frequencies of digits do not exist.