A Study on the General M-J Set of Rational Functions with One Parameter

In this paper,general Mandelbrot and Julia sets of rational functions with one parameter based on Newton's method are discussed.Firstly,the algorithms to construct their images are presented.Then,the bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of rational functions are provided.Finally,the similarity between general Mandelbrot sets and Julia sets,and that between common Mandelbrot sets and Julia sets of zⁿ+c are investigated.Experimental results show that those two similarity relationship are analogous.This result can explain the fact that the dynamics on the complex plane have the close connection with the dynamics on the parameter plane,as well as the universality of these connections,and hence produce promising directions for improving the theory of Mandelbrot and Julia sets.