Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cubic polynomials with a 2-cycle of Siegel disks

Yuming Fu, Jun Hu, Oleg Muzician

Abstract

Under conjugation by affine transformations, the dynamical moduli space of cubic polynomials $f$ with a $2$-cycle of Siegel disks is parameterized by a three-punctured complex plane as a degree-$2$ cover. Assuming the rotation number of $f^2$ on the Siegel disk is of bounded type, we show that on the three-punctured complex plane, the locus of the cubic polynomials with both finite critical points on the boundaries of the Siegel disks on the $2$-cycle is comprised of two arcs, corresponding to the cases with two critical points on the boundary of the same Siegel disk, and a Jordan curve, corresponding to the cases with two critical points on the boundaries of different Siegel disks.

Cubic polynomials with a 2-cycle of Siegel disks

Abstract

Under conjugation by affine transformations, the dynamical moduli space of cubic polynomials with a -cycle of Siegel disks is parameterized by a three-punctured complex plane as a degree- cover. Assuming the rotation number of on the Siegel disk is of bounded type, we show that on the three-punctured complex plane, the locus of the cubic polynomials with both finite critical points on the boundaries of the Siegel disks on the -cycle is comprised of two arcs, corresponding to the cases with two critical points on the boundary of the same Siegel disk, and a Jordan curve, corresponding to the cases with two critical points on the boundaries of different Siegel disks.

Paper Structure

This paper contains 17 sections, 30 theorems, 50 equations, 24 figures.

Table of Contents

  1. Introduction
  2. The parameter space $\Sigma_\theta$ and Dynamics of $f_{\alpha}$ when $\alpha\in\Sigma_{\theta}$
  3. The form of $f_\alpha$
  4. Critical points of $f_\alpha$
  5. Julia set $J(f_\alpha)$ when $\alpha \notin \partial \mathcal{M}$
  6. Boundary of the connectedness locus $\mathcal{M}_\theta$
  7. Boundaries of Siegel disks
  8. Combinatorics of the Fatou components of $f_\alpha$ when $\alpha\in\Gamma_\theta\cup\Gamma_\theta^0\cup\Gamma_\theta^1$
  9. External rays landing a separating repelling fixed point
  10. Labels of eventually Siegel-disk components when $\alpha\in \Gamma_\theta$
  11. Labels of eventually Siegel-disk components when $\alpha\in \Gamma_{\theta}^0\cup \Gamma_{\theta}^1$
  12. Proof of the Main Theorem
  13. Lebesgue measure of $J(f_\alpha)$
  14. Quasiconformal extensions to multiply connected domains
  15. Conformal angle, and rigidity of combinatorial pattern
  16. ...and 2 more sections

Key Result

Proposition 2.1

Let $f$ be a cubic polynomial having a $2$-cycle of Siegel disks with rotation number $\theta$. Then $f$ is conformally conjugate to $f_\alpha$ in the form of (f-alpha-family) for some $\alpha\in\mathbb{C}\setminus\{0,-1\pm \sqrt{1-\lambda}\}$. In particular, $f_\alpha$ has a $2$-cycle of Siegel dis

Figures (24)

  • Figure 1: The parameter plane of $\Sigma_\theta$ with $\theta=(\sqrt{5}-1)/2$.
  • Figure 2: The Julia set of $f_\alpha$ for $\alpha=-2-0.2i$, a disconnected case.
  • Figure 3: The Julia set of $f_\alpha$ for $\alpha=-2.5+2.5i$, a hyperbolic-like case.
  • Figure 4: The Julia set of $f_\alpha$ for $\alpha=1.6066+.41583i$, a capture case.
  • Figure 5: All possible landing patterns for periodic external rays of period $2$.
  • ...and 19 more figures

Theorems & Definitions (57)

  • Definition
  • Remark 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5: Zha11
  • Corollary 2.6
  • ...and 47 more