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Polynomials in molecules

Yan Gao, Jinsong Zeng

TL;DR

The paper develops a comprehensive framework connecting the combinatorics of Fatou chains, laminations, and perturbation theory to the parameter-space structure of polynomials. It proves that geometrically finite polynomials lie in molecules exactly when each critical point lies in a maximal Fatou chain, and that distinct molecules are disjoint, with subhyperbolic maps characterized by their relation to maximal Fatou chains. A dynamical perturbation technology moves last critical points from the Julia set into Fatou domains, enabling hyperbolic–subhyperbolic deformations and semiconjugacies that preserve key combinatorial data. The authors extend the molecule concept via extended molecules, show equivalence with the original notion, and completely characterize which subhyperbolic maps lie on hyperbolic boundaries in terms of Fatou-chain dynamics. The results synthesize hyperbolic-parabolic deformation with a novel dynamical perturbation approach to illuminate the global organization of polynomial parameter space in terms of molecules and laminations.

Abstract

This paper characterizes polynomials within molecules. We show that a geometrically finite polynomial of degree $d\geq2$ lies in a molecule if and only if all its critical points belong to maximal Fatou chains, and show that distinct molecules are mutually disjoint. We also establish a necessary and sufficient condition for subhyperbolic polynomials to be on the closures of bounded hyperbolic components.

Polynomials in molecules

TL;DR

The paper develops a comprehensive framework connecting the combinatorics of Fatou chains, laminations, and perturbation theory to the parameter-space structure of polynomials. It proves that geometrically finite polynomials lie in molecules exactly when each critical point lies in a maximal Fatou chain, and that distinct molecules are disjoint, with subhyperbolic maps characterized by their relation to maximal Fatou chains. A dynamical perturbation technology moves last critical points from the Julia set into Fatou domains, enabling hyperbolic–subhyperbolic deformations and semiconjugacies that preserve key combinatorial data. The authors extend the molecule concept via extended molecules, show equivalence with the original notion, and completely characterize which subhyperbolic maps lie on hyperbolic boundaries in terms of Fatou-chain dynamics. The results synthesize hyperbolic-parabolic deformation with a novel dynamical perturbation approach to illuminate the global organization of polynomial parameter space in terms of molecules and laminations.

Abstract

This paper characterizes polynomials within molecules. We show that a geometrically finite polynomial of degree lies in a molecule if and only if all its critical points belong to maximal Fatou chains, and show that distinct molecules are mutually disjoint. We also establish a necessary and sufficient condition for subhyperbolic polynomials to be on the closures of bounded hyperbolic components.
Paper Structure (13 sections, 37 theorems, 43 equations, 2 figures)

This paper contains 13 sections, 37 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fatou chains of polynomials
  3. Dynamical Perturbation of polynomials
  4. Parabolic points and convergence of external rays
  5. Dynamical Perturbation Theorem
  6. Proof of Theorem \ref{['thm:perturbation']}
  7. Lamination and geodesic lamination
  8. The extended molecules
  9. Hyperbolic-parabolic deformation
  10. Reduction along relative hyperbolic components
  11. Geometrically finite maps in extended molecules
  12. Extended molecules are molecules
  13. When is a subhyperbolic map on the boundary?

Key Result

Corollary 1.2

For every degree $d\geq 2$, we have $\mathscr M(d)={\mathscr M}_+(d)$.

Figures (2)

  • Figure 1: The construction of $F_n$ on $D_n(c^*)$ and $D_n(a_j)$.
  • Figure 2: Illustration of Proposition \ref{['pro:plumbing2']}.

Theorems & Definitions (71)

  • Definition 1.1
  • Corollary 1.2
  • Lemma 2.1: BOPT, Theorem B
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 61 more