Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn

Abstract

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.