The generalized Mandelbrot-Julia sets from a class of complex exponential map (Q856050)

The authors generalize some works by \textit{I. N. Baker} and \textit{P. J. Rippon} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 9, 49--77 (1984; Zbl 0558.30029)], \textit{R. L. Devaney} [Bull. Am. Math. Soc., New Ser. 11, 167--171 (1984; Zbl 0542.58021)] and \textit{M. Romera, G. Pastor} and \textit{G. Alvarez} [Growth complex exponential dynamics. Computers and Graphics 24, 115--131 (2000)] and construct a series of generalized Mandelbrot-Julia (M-J) sets from the complex exponential map. Using the experimental mathematics method, they innovate as follows: (1) They present the theoretical proof about the explosion of the generalized J sets for a complex index number; (2) they theoretically analyze the symmetry and periodicity of the generalized M-J sets; (3) they present a new attaching rule that describes the distribution of periodicity petal of generalized M sets for complex index number; (4) they find abundant structure information of the generalized J sets contained in the generalized M sets for a complex index number; (5) they 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 a real index number, a parameter value 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.