Fractal Structures of General Mandelbrot Sets and Julia Sets Generated From Complex Non-Analytic Iteration F_m (z) = z~m + C

摘要

In this paper we use the same idea as the complex analytic dynamics to study general Mandelbrot sets and Julia sets generated from the complex non-analytic iteration. The definition of the general critical point is given, which is of vital importance to the complex non-analytic dynamics. The general Mandelbrot set is proved to be bounded, axial symmetry by real axis, and have (m+1)-fold rotational symmetry. The stability condition of periodic orbits and the boundary curve of stability region of one-cycle are given. And the general Mandelbrot sets are constructed by the escape-time method and the periodic scanning algorithm, which present a better understanding of the structure of the Mandelbrot sets. The filled-in Julia sets Km,c have m-fold structures. Similar to the complex analytic dynamics, the general Mandelbrot sets are kinds of mathematical dictionary or atlas that map out the behavior of the filled-in Julia sets for different values of c.