The Study Of The M-J Sets Of Complex Quadratic Mappings

In this paper mainly studies the connections of the Mandelbrot sets and Julia sets of the complex quadratic mappings family of and the Mandelbrot sets and Julia sets of the complex family of quadratic mappings f : zâ†'z ~2+ c ( c∈C)which we have known. Under the conditions with the different of parameter, gives the corresponding relationship of Mandelbrot sets and Julia sets of two types of complex mappings family. First of all,this paper gives the cladograms of the special types of family of complex quadraticmappings g :wâ†'aw ~2+ e, g : wâ†'( w -μ)~2+ e, g : wâ†'a ( w -μ)~2+ e and the family of complex quadratic mappings f : zâ†'z ~2+ c ( c∈C),and from the cladogram obtains that the branch points and bifurcation between the complex quadratic mappings family is similar, and knows that these is a link between these complex quadratic mappings family, Next, respectively, discusses the connections of the Mandelbrot sets and Julia sets of the general family of these complex quadratic mappings,and the Mandelbrot sets and Julia sets of the complex family of quadratic mappings f : zâ†'z ~2+ c ( c∈C) which we have known. The research shows that we can find one continuous mappings for each condition. With these mappings, we can establish the corresponding relationship between the Mandelbrot sets and Julia sets. Finally, on the basis of the previous, in this paper discusses the general family of complex quadratic mappings, and gives two continuous mappings, gets the connections of the Mandelbrot sets and Julia sets of the general family of complex quadratic mappings g :wâ†'aw ~2+ bw + e and the Mandelbrot sets and Julia sets of the complex family of quadratic mappings f : zâ†'z ~2+ c ( c∈C) which we have known. The paper also gives the images of the Mandelbrot sets and Julia sets of these complex quadratic mappings family with the corresponding initial value. With the images, this paper tests the correctness of the given theorem.