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The high type quadratic Siegel disks are Jordan domains

Mitsuhiro Shishikura, Fei Yang

Abstract

Let $α$ be an irrational number of sufficiently high type and suppose $P_α(z)=e^{2πiα}z+z^2$ has a Siegel disk $Δ_α$ centered at the origin. We prove that the boundary of $Δ_α$ is a Jordan curve, and that it contains the critical point $-e^{2πiα}/2$ if and only if $α$ is a Herman number.

The high type quadratic Siegel disks are Jordan domains

Abstract

Let be an irrational number of sufficiently high type and suppose has a Siegel disk centered at the origin. We prove that the boundary of is a Jordan curve, and that it contains the critical point if and only if is a Herman number.

Paper Structure

This paper contains 27 sections, 44 theorems, 279 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Topology and obstructions of Siegel disk boundaries
  3. The statement of the main result
  4. Strategy of the proof
  5. Some observations
  6. Near-parabolic renormalization scheme
  7. Inou-Shishikura's class
  8. Near-parabolic renormalization
  9. Some sets in the Fatou coordinate planes
  10. Some quantitative estimates
  11. Renormalization tower and orbit relations
  12. The suitable heights
  13. Radii of Siegel disks
  14. Definition of the heights
  15. The location of the neighborhoods
  16. ...and 12 more sections

Key Result

Theorem 2.1

For all $f\in\mathcal{IS}_0\cup\{Q_0\}$, there exist two simply connected domains $\mathcal{P}_{attr,f}$, $\mathcal{P}_{rep,f}\subset U_f$ and two univalent maps $\Phi_{attr,f}:\mathcal{P}_{attr,f}\to\mathbb{C}$, $\Phi_{rep,f}:\mathcal{P}_{rep,f}\to\mathbb{C}$ such that

Figures (8)

  • Figure 1: The domains $U$ (the gray part), $U'$ (the white region bounded by the blue curves, see \ref{['equ-U-pri']} for the definition) and their successive zooms near $-1$. The outer boundary of $U'$ looks like a circle with radius about $35$ and the rightmost point of $U$ is about $32.2$. The widths of these pictures are $72$, $0.6$ and $0.0075$ respectively. It can be seen clearly from these pictures that $\overline{U}\cap(-\infty,-1]=\emptyset$ and $U\Subset U'$.
  • Figure 2: The perturbed Fatou coordinate $\Phi_f$ and its domain of definition $\mathcal{P}_f$. The image of $\mathcal{P}_f$ under $\Phi_f$ has been colored accordingly by the same color on the right. The blue set on the left depicts the forward orbit of the critical point $\textup{cp}_f$.
  • Figure 3: Left: The sets $\mathcal{C}_f$, $\mathcal{C}_f^\sharp$ and some of their preimages. The blue set depicts the forward orbit of the critical point $\textup{cp}_f$. Right: The images of $\mathcal{C}_f\cup\mathcal{C}_f^\sharp$ and $S_f$ under the perturbed Fatou coordinate $\Phi_f$ and it shows how the near-parabolic renormalization map is induced.
  • Figure 4: The inverse $\Phi_f^{-1}$ of the perturbed Fatou coordinate can be extended holomorphically to $\widetilde{\mathcal{D}}_f$ (colored cyan). It can be seen that the image $\Phi_f^{-1}(\widetilde{\mathcal{D}}_f)$ wraps around $0$. The holomorphic map $\Phi_f^{-1}$ has an anti-holomorphic lift $\chi_f$ such that $\mathbb{E}\textup{xp}\circ\chi_f=\Phi_f^{-1}$ (note that $\mathbb{E}\textup{xp}$ is anti-holomorphic). Some special points are also marked.
  • Figure 5: In the dynamical plane of $f_n$, the sets $\partial^l V_n^0$, $\partial^r V_n^0$ and $I_n^0$ are colored cyan, purple and red respectively. The blue set depicts the (partial) forward orbit of the critical point $\textup{cp}_{f_n}$. The sets $V_n^0$ and $\widetilde{\mathcal{C}}_n^\sharp=f_n^{\circ k_n}(V_n^0)$ are colored gray.
  • ...and 3 more figures

Theorems & Definitions (90)

  • Definition : Herman numbers
  • Theorem 2.1: Leau-Fatou Mil06 and Inou-Shishikura IS08
  • Definition : Neighborhoods of a function
  • Proposition 2.2: BC12, see Figure \ref{['Fig_perturbed-Fatou-coor']}
  • Definition : see Figure \ref{['Fig_near-para-norm-defi']}
  • Proposition 2.3: Che19, see Figure \ref{['Fig_near-para-norm-defi']}
  • Definition : Near-parabolic renormalization, see Figure \ref{['Fig_near-para-norm-defi']}
  • Theorem 2.4: IS08
  • Lemma 2.5
  • proof
  • ...and 80 more