Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gluing polynomials along the circle

Panjing Wu, Gaofei Zhang

TL;DR

The paper develops a framework to glue two polynomials along the Jordan boundary of their marked immediate basins and proves that, for polynomials with connected Julia sets and a shared finite super-attracting fixed point of degree $d_0$ that satisfy an independence condition, there exists a unique rational map $G$ realizing the simultaneous uniformization of the pair. The core method combines puzzle theory, a Closing Lemma that enables hyperbolic approximations, and Thurston-type rigidity to pass from a topological gluing to a geometric one, yielding a continuous gluing operator $\mathcal{G}$ on the non-renormalizable class. A compactness argument shows the limit of gluings for hyperbolic approximants preserves the puzzle structure, and non-renormalizable holomorphic dynamics provides shrinking controls and rigidity to identify the limit map uniquely with $G$. The results extend the realm of simultaneous uniformization beyond PCF polynomials and establish a robust mechanism for realizing topological gluings as actual rational maps, with implications for mating-type constructions and deformation theories in holomorphic dynamics.

Abstract

Gluing is a cut and paste construction where the dynamics of a map in a given domain is replaced by a different one, under the condition that the two agree along the gluing curve. Here we consider two polynomials with a finite super-attracting fixed point of the same degree. We prove that any two such non-renormalizable polynomials can be glued into a rational map along the Jordan boundary of the immediate basin of the super-attracting fixed point.

Gluing polynomials along the circle

TL;DR

The paper develops a framework to glue two polynomials along the Jordan boundary of their marked immediate basins and proves that, for polynomials with connected Julia sets and a shared finite super-attracting fixed point of degree that satisfy an independence condition, there exists a unique rational map realizing the simultaneous uniformization of the pair. The core method combines puzzle theory, a Closing Lemma that enables hyperbolic approximations, and Thurston-type rigidity to pass from a topological gluing to a geometric one, yielding a continuous gluing operator on the non-renormalizable class. A compactness argument shows the limit of gluings for hyperbolic approximants preserves the puzzle structure, and non-renormalizable holomorphic dynamics provides shrinking controls and rigidity to identify the limit map uniquely with . The results extend the realm of simultaneous uniformization beyond PCF polynomials and establish a robust mechanism for realizing topological gluings as actual rational maps, with implications for mating-type constructions and deformation theories in holomorphic dynamics.

Abstract

Gluing is a cut and paste construction where the dynamics of a map in a given domain is replaced by a different one, under the condition that the two agree along the gluing curve. Here we consider two polynomials with a finite super-attracting fixed point of the same degree. We prove that any two such non-renormalizable polynomials can be glued into a rational map along the Jordan boundary of the immediate basin of the super-attracting fixed point.

Paper Structure

This paper contains 19 sections, 8 theorems, 54 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Examples of the gluing
  4. Strategy of the proof
  5. The gluing operator $\mathcal{G}$ for PCF polynomials
  6. Puzzle structure for the topological gluing
  7. Puzzle construction for polynomials
  8. Puzzle construction for topological gluing
  9. The closing Lemma
  10. Proof of the Closing Lemma
  11. Handling the cluing curve
  12. Post-critically finite approximation
  13. Compactness argument
  14. Non-renormalizable holomorphic dynamics
  15. Uniform control of the puzzle size
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $f \in\mathcal{P}_{d_1, d_0}$ and $g \in\mathcal{P}_{d_2, d_0}$ be two post-critically finite polynomials satisfying the independence condition. Then they can be glued into a rational map $G \in \mathcal{R}_{d_1 + d_2 - d_0}$.

Figures (15)

  • Figure 1: The Julia set for $z^3 \frac{z - 3}{1 - 3z}$ with $1$ being a parabolic fixed point having two petals which are symmetric about the unit circle.
  • Figure 2: The Julia set for $e^{2\pi i \theta_0} z^3 \frac{z - 2}{1 - 2z}$ with a fixed parabolic petal which is symmetric about the unit circle. The Julia set for $e^{-2\pi i \theta_0} z^3 \frac{z - 2}{1 - 2z}$ is its symmetric image about the real line.
  • Figure 3: The Julia set for $f_\alpha$.
  • Figure 4: The Julia set for $f_\beta$.
  • Figure 5: The Julia set for $\mathcal{G}(f_\alpha, f_\beta)$. When viewed from infinity and the origin, one sees respectively the Julia sets for $f_\alpha$ and $f_\beta$.
  • ...and 10 more figures

Theorems & Definitions (12)

  • Theorem 2.1: Zhang, Zhang
  • Definition 1
  • Lemma 3.1
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • Theorem 6.1
  • ...and 2 more