Quasisymmetries of finitely ramified Julia sets

TL;DR

This paper explores quasisymmetries in fractal structures called Julia sets, showing how they can be used to understand and classify these complex shapes.

Contribution

The paper introduces a new theory of quasisymmetries for finitely ramified Julia sets and generalizes previous results on their symmetry groups.

Findings

Certain finitely ramified fractals have undistorted metrics that are quasisymmetrically equivalent.

Connected Julia sets for hyperbolic unicritical polynomials have infinitely many quasisymmetries.

Quasisymmetry groups of some Julia sets include Thompson’s group F.

Abstract

We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of “undistorted metrics” that are all quasisymmetrically equivalent. As a result, piecewise-defined homeomorphisms of such a fractal that locally preserve the cell structure are quasisymmetries. This immediately gives a solution to the quasisymmetric uniformization problem for topologically rigid fractals such as the Sierpiński triangle. We show that our theory applies to many finitely ramified Julia sets, and we prove that any connected Julia set for a hyperbolic unicritical polynomial has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove that the quasisymmetry group of the Julia set for the rational function \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1-z^{-2}$$\end{document} is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson’s group F.

Figures