Kovalenko V. Analytic methods in fractal geometry

In the thesis, we develop analytic methods for the study of fractal objects on the complex plane: curves, sets of incomplete sums of absolutely convergent series with complex terms, continuous non-dierentiable mappings, probability measures with fractal supports. To express and represent numbers in numeral system with even positive integer base and symmetric minimal redundant alphabet we introduce pragmatic representation with zero redundancy. Using geometry of this representation we construct an analytic representation of the Koch snowflake as the image of a certain complex valued function of real argument. An explicit analytic expression for homeomorphism of the circle to the Koch snowake and one-parameter family of fractal curves containing the Koch snowake are found. The structural, topological, metric and self-similar properties of these curves are studied. We describe the class of sets generalizing the sets of incomplete sums of absolutely convergent series with complex terms. Sucient conditions for these sets to be disconnected or connected, to have interior points, to be of zero two-dimensional Lebesgue measure are found. Hausdorff-Besicovitch dimension of these sets is also estimated. Vicsek fractal is a fractal curve of web type. We study the structural, topological, metric and fractal properties of probability distributions on the Vicsek fractal induced by the discrete distributions of symbols of encoding of curve's points with the 5-symbol alphabet.