Newton法对应单参有理函数族的广义M-J集

讨论了Newton法对应单参数有理函数族的广义Mandelbrot集和Julia集,给出了它们的构造算法,证明了其广义Mandelbrot集的有界性,并给出了其周期点个数的计算公式.利用数学实验的方法研究了广义Mandelbrot集周期芽苞分布规律,并通过对比分析得到了它们与z~n+c的Mandelbrot集和Julia集之间的族相似性类似的新的族相似关系.文中算法为Mandelbrot集和Julia集的发展提供了新的思路.

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 z~n+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.