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

Abstract

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.

Introduction

The experimental mathematics method utilizing computer to develop research work in fractal, which has become an independent research direction [1], [2]. Through utilizing the ability of the powerful calculation and drawing of computers, the researchers expanded the experiment and research in the fractal fields, and did some theoretic analysis based on the experiments. They have found that regular structures contain in the fractal sets, thereby enriched the fractal theory [3], [4], [5], [6], [7], [8]. In 1981, the dynamics behavior of complex exponential map z ← e^z, was studied for the first time by Misiurewicz [9]. After this, the generalized M–J sets generated from complex exponential map z ← e^λz, z ← λe^z, $z\leftarrow {\mathrm{e}}^{z^2+\lambda }$ and z ← e^z/λ (λ ∈ C), was independently constructed and studied by Baker and coworkers [10], [11], [12], [13], [14], [15]. In this paper, we shall develop the work of Baker, Devaney and Romera, etc. It is easy to show the complex maps z ← e^λz and z ← λe^z are equivalence [13], so we shall only research the fractal structure of the generalized M–J sets generated from three complex exponential maps:

$$f_1:z\leftarrow \lambda {\mathrm{e}}^{z^w}(\lambda \text{,}w\in C)\text{,}$$

$$f_2:z\leftarrow {\mathrm{e}}^{z^w+\lambda }(\lambda \text{,}w\in C)\text{,}$$

$$f_3:z\leftarrow {\mathrm{e}}^{z^w/\lambda }(\lambda \text{,}w\in C)\text{.}$$