Fractal structures of the non-boundary region of the generalized Mandelbrot set

In order to study the interior structure of the Mandelbrot(M) set, Pickover and Hooper have advanced the epsilon cross method and the star trails method, respectively. In order to reveal the exterior structure of the M set, Philip has proposed the regional decomposition method and the angle-slicing decomposition method. In this paper, these methods are extended, and the equipotential line method and the color adjustment method are proposed to study the exterior structure of the M set..4 series of the non-boundary region structure images of the generalized M sets is generated by using the above-mentioned methods. The results show that the non-boundary regions of the generalized M sets for integer index number have the fractal feature; while those for decimal index number have discontinuity and collapse. and their evolutions depend on the choice of the principal range of the phase angle.