Mandelbrot set, which was provided as a highlight in fractal and chaos, is studied by many researchers. With the extension of Mandelbrot set to generalized $M$ set with different kinds of exponent $k$ ($k-M$ set), properties are hard to understand when $k$ is a complex number. In this paper, fractal property of generalized $M$ set with complex exponent $z$ is studied. First, a relation is constructed between generalized $M$ set with complex and real exponent. Then, distribution of $z-M$ set on complex plane is researched. Meanwhile, symmetry of generalized $M$ set is proved. Finally, graphics, generated by escape time algorithm, are the validated results of this paper.

中文翻译：

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具有复指数的广义曼德尔布罗特集的分形性质

曼德尔布罗特集是分形和混沌领域的一个亮点，受到许多研究人员的研究。将Mandelbrot集扩展为具有不同类型指数 $k$ （ $k-M$ 集）的广义 $M$ 集，当 $k$ 的广义 $M$ 集的分形性质。首先，在具有复指数和实指数的广义 $M$ 集合之间构造关系。然后研究了 $z-M$ 集合在复平面上的分布。同时证明了广义 $M$ 集合的对称性。最后，利用逃逸时间算法生成的图形是本文的验证结果。