General Mandelbrot Sets and Julia Sets Generated from Non-analytic Complex Iteration F_m(z)=z^n+c

Abstract

In this paper we investigate the general Mandelbrot sets and Julia sets generated from non-analytic complex iteration . We use the escaping time algorithm and the periodic scanning algorithm to construct the general Mandelbrot set and its local enlargement. We find that general Mandelbrot sets are symmetrical by reflection in the real axis, and has (m+1)- fold rotational symmetry around 0, the Julia sets have m-fold structures. Similar to the analytic complex iterated function systems, the general Mandelbrot sets are kinds of mathematical dictionary or atlas that map out the behavior of the Julia sets for different values of c.