Quasisymmetries of finitely ramified Julia sets

TL;DR

This paper develops a theory of quasisymmetries for finitely ramified fractals, applies it to Julia sets, and reveals the rich structure of their quasisymmetry groups, including infinite groups and connections to Thompson's group F.

Contribution

It introduces a framework for understanding quasisymmetries of finitely ramified Julia sets and solves the uniformization problem for certain fractals, extending previous results.

Findings

Finitely ramified fractals admit quasi-equivalent undistorted metrics.

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

The quasisymmetry group of the Julia set for 1-z^{-2} is infinite and contains 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 quasi-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 Sierpi\'nski 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 $1-z^{-2}$ is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson's group $F$.