We present a simple proof of Tan’s theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman’s lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan’s theorem.

中文翻译：

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Zalcman函数以及Mandelbrot集，Julia集和Tricorn之间的相似性

我们给出了在Misiurewicz参数上的Mandelbrot集和Julia集之间的渐近相似性的Tan定理的简单证明。然后，我们根据Zalcman函数，即通过将Zalcman引理应用于复杂动力学生成的整个函数，对这种现象给出新的观点。我们还显示了Misiurewicz参数上的Tricorn和Julia集之间的渐近相似性，这是Tan定理的反全同形式。