Computer Construction And Research On Several Types Of Fractal Images

This paper begins with the fundamental theory, discusses the theory of Julia sets and its relative application. The main contents as follows:(1) The Julia sets of Newton's method is a fascinating problem in the study of the fractals. The studies about basins of attraction and inner structures of Julia sets with Newton's method are helpful to understand the essence of approximating the roots with iterative methods. This paper constructs fractal images for the standard's Newton method, Halley's method, and Schroder's method, analyzes the Julia sets theory of them, studies structural characteristics of Julia sets, shows the properties and conditions of fixed points for the standard Newton method, Halley's method, and Schroder's method. This paper also observes the relation between roots of polynomial and Julia sets structure, namely, if keeping the relative position of the roots of a polynomial invariable, then the topological structure is also invariable. If there is an extraneous fixed point, then the extraneous fixed point is also invariable. Otherwise, the structures of Julia sets and fixed points would be changed.(2) The paper applies the fractal theory to some exponential equations, studies the relation between roots of some complex exponential equation and theory of Julia sets. The paper extendes Kim's complex exponential function, comes up with theory about Julia sets of Newton's transformation for general exponential equation, analyzes the behavior of the roots of some complex exponential equation, and proves the Julia set's symmetry, boundedness and embedding topology distribution structure of basins of attraction in theory.(3) Collatz put forward 3x + 1 problem at first in an international mathematics convention, hence it is known as "Collatz problem". International mathematics world have studied 3x + 1 problem deeply for fifty years and put forward the several conjectures. This paper generalizes 3x + 1 function to the complex plane, gains two different complex maps. Then the paper constructs fractal images for this two complex maps using escape time, stopping time and total stopping time arithmetic respectively, studies the dynamics for generalized 3x +1 function on the base of the structure characteristics of the fractal images. The fractal images constructed by the three arithmetic shows that there are complicated fractal structure characteristics for 3x + 1 function in the complex plane. It states that 3x + 1 function has stable convergence by comparing the fractal images.