摘要
In this paper we study the global structure of periodic orbits for a one-dimensionalcomplex map Z_(n+1) =Z_n^m+C by the algebraic analytical method advanced by the authot in1985. We give a general formula for the calculation of the orbit number H_N of any period--Norbit. We also verify rigorously that the complex structures of the Mandelbrot set (m =2)and its generalized form (m>2) are composed of infinitely many stable regions of differentperiodic orbits. We find out that the relation between the stable region number I_N of theperiod-N orbit and its orbit number H_N is exactly I_N =N·H_N/m. The algebraic equstionsof the boundary of each element and the locations of its cusp and center can be given pre-cisely. Furthermore, the cause A the infinitely nested structures for these complex figures areexplained.
In this paper we study the global structure of periodic orbits for a one-dimensionalcomplex map Z<sub>n+1</sub> =Z<sub>n</sub><sup>m</sup>+C by the algebraic analytical method advanced by the authot in1985. We give a general formula for the calculation of the orbit number H<sub>N</sub> of any period--Norbit. We also verify rigorously that the complex structures of the Mandelbrot set （m =2）and its generalized form （m>2） are composed of infinitely many stable regions of differentperiodic orbits. We find out that the relation between the stable region number I<sub>N</sub> of theperiod-N orbit and its orbit number H<sub>N</sub> is exactly I<sub>N</sub> =N·H<sub>N</sub>/m. The algebraic equstionsof the boundary of each element and the locations of its cusp and center can be given pre-cisely. Furthermore, the cause A the infinitely nested structures for these complex figures areexplained.
