The generalized M sets from a class complex mapping system

摘要

The method constructing the generalized Mandelbrot sets (the generalized M sets) for positive integer index number generated from the composition of two simple complex mapping which was put forward by Shirriff was expanded. Based on a class simple complex mapping system expanded by the author, a series of the generalized M sets for real index number were constructed. Using the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, the structural characteristics and evolutions of the generalized M sets were studied. The results show that the generalized M sets for integer index number have symmetry and fractal feature; while the generalized M sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle. And the author sets forth the physical meaning for the generalized M sets.