Global analytical structure of the Mandelbrot set and its generalization (Q1201592)

The author studies the global structure of periodic orbits for a one- dimensional map \(Z_{n+1}=Z^ m_ n+C\) (\(C\) complex, \(m\geq 2\)) using an earlier developed algebraic analytical method. He gives a general formula for the calculation of the orbit number of any orbit with finite period. He verifies that the complex structures of the Mandelbrot set (\(m=2\)) and its generalized form (\(m>2\)) are composed of infinitely many stable regions of different periodic orbits. A simple linear relation between the stable region number of the periodic orbit and its orbit number is derived. The algebraic equations of the boundary of each element and the locations of its cusp and the center are given precisely and the cause for the infinitely nested structures in these complex figures is explained.