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The general quaternionic M-J sets on the mapping \(z\leftarrow z^{\alpha}+c \, (\alpha \in \mathbf N)\). (English) Zbl 1152.28321

Summary: Fractal nature exists not only in complex plane but also in higher dimensional space. It is a focus to find new modals to construct new 3-D fractals. In this paper, the general Mandelbrot sets and Julia sets on the mapping \(z\leftarrow z^{\alpha}+c \, (\alpha \in \mathbf N)\) are discussed. The 3-D projections of M-J sets are constructed using escape time algorithm and ray-tracing method. And their properties are theoretically analysed. The connectness of the general quaternionic M sets is proved and the boundary of the stability region of the fixed point is calculated. It is found that if the parameter c of Julia sets is chosen from the M sets, they share the same cycle number and stable points. It can be concluded that the quaternionic M sets contain sufficient information of quaternionic Julia sets.

MSC:

28A80 Fractals
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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