Collet-Eckmann type conditions and conformal welding of unicritical quadratic laminations

TL;DR

The paper develops a combinatorial Collet-Eckmann (CCE) condition for unicritical laminations on the unit circle and proves that CCE implies a Hölder continuous conformal welding that reconstructs a Julia set for a unicritical polynomial. It shows that for degree two, almost every angle satisfies CCE, which yields Julia sets with Hölder Fatou components without relying on Beurling’s theorem, and it outlines a general framework via generalized cylinder sets and digit-fixing to handle the combinatorics. The approach connects lamination dynamics with Julia-set geometry through a symbolic-dynamical toolkit (itineraries, gluing links, pullbacks) and leverages the Riemann mapping to translate geometric properties into combinatorial conditions. The results provide a generic mechanism ensuring Hölder regularity in unicritical polynomial Julia sets and offer a robust method for establishing Hölder conformance and removability properties in this setting.

Abstract

In this paper, we introduce a Collet-Eckmann type condition for the unicritical laminations on the unit circle. We prove that this condition implies the lamination admits a Hölder continuous conformal welding which produces a Julia set for some unicritical polynomial. In consequence, we present a new proof that almost all angles on the unit circle produce quadratic polynomials with Hölder Fatou components, without the use of Beurling's theorem.

Figures (10)

• Figure 1: Quadratic lamination $\approx^{(2)}_\alpha$ with $\alpha \approx e^{0.35\pi i}$ and its corresponding quadratic Julia set $J_c$ with $c\approx-0.326+0.102i$. Riemann map $\varphi_c$ maps $\alpha$ to $c$ and "glues" $\approx_\alpha^{(2)}$.
• Figure 2: Outline for the proof of Theorem \ref{['main_theo']}
• Figure 3: The cylinder sets for $\alpha=e^{4\pi i/7}$ associated with words of length $3$ with $d=2$.
• Figure 4: Picture for Proposition \ref{['gluing_links']} where $d=2$. The gluing link $D_1$ contains $\alpha$ and thus its preimage has one component, whereas the gluing link $D_2$'s preimage has two separate gluing links as it does not contain $\alpha$.
• Figure 5: A nice gluing circuit around $x$

Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: thurston2020degree
• Lemma 2.2: lin2019conformal
• Definition 2.3: Itinerary Maps, lin2018quasisymmetry
• Lemma 2.4: bandt2006symbolicthurston2020degree
• Definition 2.5: Cylinder Sets, lin2018quasisymmetry
• Definition 2.6: Gluing links
• Proposition 2.7: see Figure \ref{['fig:links']}
• Lemma 2.8