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Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies

Tomoki Kawahira

Abstract

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and investigation on the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia sets.

Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies

Abstract

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and investigation on the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia sets.

Paper Structure

This paper contains 24 sections, 26 theorems, 49 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Degeneration pair and degenerating arc system
  3. External and internal landing
  4. Degenerating arcs.
  5. Degenerating arc system
  6. Critical sectors
  7. Tessellation
  8. Fundamental model of tessellation
  9. Tessellation for $\boldsymbol{f}$ and $\boldsymbol{g}$.
  10. Edge sharing
  11. Tiles and panels with small diameters
  12. Pinching semiconjugacy
  13. Proof of Theorem \ref{['thm_semiconj']}
  14. Continuity of pinching semiconjugacies
  15. Proof of (i)
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(f \to g)$ be a degeneration pair. There exist families $\mathrm{Tess}(f)$ and $\mathrm{Tess}(g)$ of simply connected sets with the following properties: The properties above holds if we replace $f$ by $g$. Moreover:

Figures (13)

  • Figure 1: Chubby rabbits
  • Figure 2: Samples of tessellation. For the two figures at the upper left, parameters are taken from period 12 and 4 hyperbolic components of the Mandelbrot set as indicated in the figure of a small Mandelbrot set.
  • Figure 3: Left, the Julia set of an $f$ in segment (s2) for $p/q=1/3$, and right, one in segment (s1), with their degenerating arc system roughly drawn in. Attracting cycles are shown in heavy dots. Degenerating arcs with types ${\left\{ 1/7, 2/7/ 4/7 \right\}}$ and ${\left\{ 1/28, 23/28, 25/28 \right\}}$ are emphasized.
  • Figure 4: The fundamental model
  • Figure 5: $f^{lq}$ and $g^{lq}$ are semiconjugate to $F^q$ and $G^q$.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Theorem 1.1: Tessellation
  • Theorem 1.2: Pinching semiconjugacy
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3: Internal landing
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 16 more