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

## TL;DR

This paper provides a new proof of the asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters, introduces Zalcman functions for complex dynamics, and extends the similarity concept to the tricorn set.

## Contribution

It offers a simplified proof of Tan's theorem, introduces Zalcman functions into complex dynamics, and establishes similarity results for the tricorn set.

## Key findings

- Proof of asymptotic similarity between Mandelbrot and Julia sets
- Introduction of Zalcman functions in complex dynamics
- Similarity between the tricorn and Julia sets at Misiurewicz parameters

## 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.