Zamrii I. Fractal properties of functions associated with three-symbolic system of real numbers coding and their modifications

The thesis is devoted to researching of continuous on segment piecewise singular and singular functions defined in terms of given polybasic three-symbolic Q_3 - representation of real numbers, that depends from three parameters and is a generalization of classic ternary representation of real numbers. Also in the thesis we study for mentioned above functions their local and global properties (structural, variational, extremal, differential, integral, fractal etc). We investigate the family of continuous functions that preserve the digit 1 in Q_3 - representation of real numbers. It is proved that among the functions of this class are not exist functions with continual levels and that such functions may have no more than two infinite levels. A special role in this family has unique strictly decreasing function, called the inversor of Q_3 - representation digits, detailed description of which properties is made the thesis. Also we thoroughly study the properties of model representatives of countable subclasses of functions with one and two infinite levels respectively. We found equivalent definition for them. We put into consideration and study non-self-similar infinite-symbolic representation of real numbers, which is a modification of Q_3 - representation, describe its geometry and solve some metric problems for sets of real numbers defined by conditions for their representation.