Julia sets of the Schröder iteration functions of a class of one-parameter polynomials with high degree

In this paper the theory of Julia sets of Schröder iteration functions is introduced, the Julia sets of the Schröder functions of a one-parameter family polynomials with high degree are constructed through iteration method, and their structures are analyzed. Consequently, the following results are found in the study: (1) the Julia sets of the Schröder iteration functions of a one-parameter family polynomials with high degree contain the structure of classical Mandelbrot-like set; (2) the orbits of the critical points may escape from the zero points of the corresponding polynomial to converge to the k-cycle attractive basin or the extra fixed points; (3) if critical points on parameter plane are selected to construct Julia sets on dynamics plane, then attractive k-cycle basin will emerge, while it will not emerge if no critical points are selected; (4) the extra fixed points may be repulsive, litmusless or attractive, but the former takes the major role and (5) the Julia sets of the Schröder iteration functions have symmetry.