Quasisymmetric geometry of the Julia sets of McMullen maps

TL;DR

This paper investigates the quasisymmetric geometry of Julia sets of McMullen maps, classifying their shapes based on critical point behavior and establishing conditions under which they resemble standard fractal sets.

Contribution

It provides a comprehensive classification of Julia sets of McMullen maps based on critical point dynamics and introduces conditions for their quasisymmetric equivalence to standard fractals.

Findings

Julia sets are quasisymmetrically equivalent to Cantor sets, circles, or Sierpiński carpets.

Conditions are given for Julia sets to be Sierpiński carpets when critical points do not escape.

Most Julia sets are quasisymmetrically equivalent to round carpets, including infinitely renormalizable cases.

Abstract

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\lambda(z)=z^m+\lambda/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $\lambda\in\mathbb{C}\setminus\{0\}$. If the free critical points of $f_\lambda$ are escaped to the infinity, we prove that the Julia set $J_\lambda$ of $f_\lambda$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpi\'{n}ski carpet (which is also standard in some sense). If the free critical points are not escaped, we give a sufficient condition on $\lambda$ such that $J_\lambda$ is a Sierpi\'{n}ski carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to round carpets.