Mating Siegel and Thurston quadratic polynomials

TL;DR

The paper proves that a quadratic polynomial with a bounded-type Siegel disk and a quadratic postcritically finite polynomial are mateable. It constructs a candidate rational map $G$ modeling the mating via a Thurston-type/topological framework and shows the absence of Thurston obstructions, ensuring $G$ is the mate. A detailed homotopy-lifting construction produces a continuous semi-conjugacy $\phi$ from the formal mating to $G$, with a delicate decomposition into drops and external rays that ensures uniform convergence and a fiber structure governed by ray equivalence. Using Moore’s theorem, the quotient by ray equivalence yields a sphere, and the resulting dynamics realize the mating, extending the scope of polynomial mating to include Siegel-type dynamics coupled with postcritically finite polynomials.

Abstract

We prove that a quadratic polynomial with a bounded type Siegel disk and a quadratic post-critically finite polynomial are always mateable.

Figures (3)

• Figure 1: The construction of $T$ for $F=P_{\theta}\sqcup P_c$ with $c=-1.75488\cdots$ such that the critical point $x_0=0$ is periodic with period $3$, and $x_1=P_c(0)$, $x_2=P_c^{2}(0)$. We place the center of the Siegel disk of $F$ at infinity. The set $T$ consists of the gray regions and the black boundaries. The constructions of $U^1$ and $U^2$ for $x_1$ and $x_2$ follow Step 1, while the constructions of $U^{01}$ and $U^{02}$ for $x_0$ follow Step 2. The remaining parts are two Jordan domains $\tilde{U}^1$ and $\tilde{U}^2$.
• Figure 2: The picture on the left shows the construction of $T$ for $F=P_{\theta}\sqcup P_c$ with $c=-2$. Here, $x_2$ is a fixed point with only one joined external ray $R_{t_2}$ from $F$ landing at it, while $x_1$ is another preimage of $x_2$ under $P_c$. The picture on the right displays the construction of $T$ for $F=P_{\theta}\sqcup P_c$ with $c=-1.54368\cdots$ such that $x_1=P_c(0)$, $x_2=P^2_c(0)$, and $x_3=P^3_c(0)$ is a fixed points of $P_c$. There are multiple external rays of $P_c$ landing at each $x_i$, from which we can select one to construct the set $T$.
• Figure 3: An illustration for a ray-equivalence class of $P_{\sqrt[3]{1/4}}\sqcup P_{-2}$.

Theorems & Definitions (27)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 2.1: see Zha22
• Lemma 2.2
• proof
• Lemma 3.1: Classical shrinking lemma. LM97 (see also Man93, TY96)
• Lemma 3.2: see the Main Lemma of WYZZ23
• Definition 5: Drop-chain