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Local connectivity of Julia sets of some rational maps with Siegel disks

Shuyi Wang, Fei Yang, Gaofei Zhang, Yanhua Zhang

TL;DR

The paper addresses the local connectivity of Julia sets for rational maps with bounded-type Siegel disks by developing a puzzle-free framework. It introduces a quasi-Blaschke model G that captures exterior dynamics of f and proves a uniform contraction property for long inverse-branch compositions near the Siegel boundary, circumventing the need for puzzles. The key instrument is the Main Lemma', which controls pullbacks via (,)-admissible sequences and their contraction regions to force diameters to shrink uniformly; this underpins local connectivity via Whyburn's criterion. Applications include local connectivity for cubic Siegel polynomials along the Zakeri curve and for certain Newton and McMullen maps, extending Petersen's results to a broader class of rational maps with Siegel disks.

Abstract

We prove that a long iteration of rational maps is expanding near boundaries of bounded type Siegel disks. This leads us to extend Petersen's local connectivity result on the Julia sets of quadratic Siegel polynomials to a general case. A new key feature in the proof is that the puzzles are not used.

Local connectivity of Julia sets of some rational maps with Siegel disks

TL;DR

The paper addresses the local connectivity of Julia sets for rational maps with bounded-type Siegel disks by developing a puzzle-free framework. It introduces a quasi-Blaschke model G that captures exterior dynamics of f and proves a uniform contraction property for long inverse-branch compositions near the Siegel boundary, circumventing the need for puzzles. The key instrument is the Main Lemma', which controls pullbacks via (,)-admissible sequences and their contraction regions to force diameters to shrink uniformly; this underpins local connectivity via Whyburn's criterion. Applications include local connectivity for cubic Siegel polynomials along the Zakeri curve and for certain Newton and McMullen maps, extending Petersen's results to a broader class of rational maps with Siegel disks.

Abstract

We prove that a long iteration of rational maps is expanding near boundaries of bounded type Siegel disks. This leads us to extend Petersen's local connectivity result on the Julia sets of quadratic Siegel polynomials to a general case. A new key feature in the proof is that the puzzles are not used.

Paper Structure

This paper contains 20 sections, 28 theorems, 206 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Sketch of the proofs
  4. Quasi-Blaschke models
  5. Quasi-Blaschke models and dynamical lengths
  6. First return properties
  7. Half hyperbolic neighborhoods
  8. Contraction of $G^{-1}$
  9. Admissible sequences and contraction regions
  10. Non-critical predecessors
  11. Critical predecessors
  12. $(\delta,\eta)$-admissible sequence $\{I_n\}$
  13. Contraction regions associated to $\{I_n\}$
  14. Proof of the Main Lemma$'$
  15. Improved $(\delta,\eta)$-admissible sequences
  16. ...and 5 more sections

Key Result

Lemma 2.1

If a Jordan disk $V\subset\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}}$ contains no critical values, or its closure contains at most one critical value, then every component $U$ of $G^{-1}(V)$ is a Jordan disk. Moreover, $G:\overline{U}\to \overline{V}$ is a homeomorphism in the first case.

Figures (12)

  • Figure 1: A half hyperbolic neighborhood $H_d(I)$ of the open interval $I=(a,b)$.
  • Figure 2: The hyperbolic domains $\widetilde{\Omega}_I$, $\widehat{\Omega}_I$ and their images under the conformal map $\varphi=\varphi_3\circ\varphi_2\circ\varphi_1$.
  • Figure 3: A sketch of the local mapping relation of $G$ near the critical point $c$ of local degree $2\ell-1$ with $\ell=3$. The strictly expanding regions $U_1$ and $U_2$ of $G$ with respect to the hyperbolic metric in $\Omega$ are colored yellow.
  • Figure 4: A sketch of critical pullbacks under $G$ of types (i) and (ii). From the arc $I=(a,b)$ we obtain a new arc $J=(c,x)$.
  • Figure 5: A sketch of critical pullbacks $Q_1$, $\cdots$, $Q_{\ell-1}$ under $G$ of type (iii), which are colored yellow. A non-critical pullback $Q_\ell$ is colored cyan. From the arc $I=(a,b)$ we obtain a new arc $J=(c,x)$.
  • ...and 7 more figures

Theorems & Definitions (60)

  • Lemma 2.1
  • Definition : Dynamical length
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition : Half hyperbolic neighborhoods
  • ...and 50 more