Visualizing the complex dynamics of families of polynomials with symmetric critical points

We construct a simple complex variable polynomial mapping family fn (z) = zn + cnz which can make the origin not only be the center of an image but the boundary point of the image in the dynamic plane. We ascertain the arrangement rule of the period parameter regions in the M sets in the parameter space and determine the relationship between the image structures of the filled-in Julia sets and the parameters located in different positions in M sets. We discover that the images of the classic filled-in Julia sets from the f(z) = z2 + c are infinitely inlaid in the filled-in Julia sets from our new mapping family.