Quasisymmetries of finitely ramified Julia sets

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

Figures (17)

• Figure 1: The Sierpiński triangle has a natural finitely ramified cell structure with one $0$-cell, three $1$-cells, nine $2$-cells, and so forth.
• Figure 2: A finitely ramified cell structure for the basilica Julia set with four $1$-cells, eight $2$-cells, and sixteen $3$-cells.
• Figure 3: A finitely ramified cell structure for the bubble bath Julia set with six $1$-cells and twelve $2$-cells.
• Figure 4: The first three steps in the construction of the self-affine fractal $V(a,b,c)$.
• Figure 5: Three fractals in the $V(a,b,c)$ family, the first of which is the famous Vicsek fractal. The Euclidean metric is undistorted for the first two, but distorted for the third.

Theorems & Definitions (91)

• Theorem 1
• Theorem 2
• Definition 3
• Theorem 4
• Example 5: The basilica
• Example 6: The bubble bath
• Remark 7
• Definition 8
• Theorem 9
• Example 10: The Vicsek Family