The generalized Mandelbort–Julia sets from a class of complex exponential map

摘要

We have generalized the Baker, Devaney and Romera’s work and constructed a series of generalized Mandelbort–Julia Sets (in abbreviated form generalized M–J sets) from the complex exponential map. Using the experimental mathematics method, we have innovated as follows: Present the theoretic proof about the explosion of the generalized J sets for complex index number; Theoretically analyze the symmetry and periodicity of the generalized M–J sets; Present a new attaching rule described the distributing of periodicity petal of generalized M sets for complex index number; Find abundant structure information of the generalized J sets contained in the generalized M sets for complex index number; Find that the speed of fractal growth in generalized M–J sets for complex index number is faster than that of generalized M–J sets for real index number, the parameter value λ0 decides the speed of the fractal growth and the fractal growth in generalized M sets for complex index number points tends to the multifurcation and Misiurewicz point.