Chuikov A. Fractal analysis of functions with complex local structure defined in terms of continued fractions

The dissertation is devoted to the locally complex functions with fractal properties that are defined in terms of continued fractions (elementary and not elementary). Continuous transformations of closed interval that preserve the tails of representation of numbers by A_2-continued fractions and elementary continued fractions are built. It is constructively proved that the set of all continuous transformations of [0,5;1] that preserve tails of an A-continued fraction representation together with an operation "function composition (superposition)" form an infinite non-commutative group. For a newly defined function called quasi-inversor of the digits of the elementary continued fraction representation of numbers it is proved its nowhere monotony and cantority of set of its values. Description of the sets of its levels is obtained. Analogous of the continuous nondifferentiable functions of the Bush-Wunderlich function and the Tribin-function is constructed. In this function argument considered in the form of a nega-3-adic representation of numbers and the value of the function considered in the form of A-continued fraction representation of number. It is proved that it is continuous on [0,5;1] nowhere monotone function and has unlimited variation. The properties of the function graph and its level sets are described. Estimates of the approximation of real numbers by A_2-continued fractions are obtained. The foundations of the topological-metric theory of the representation of numbers by A_3-continued fractions are set.