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Julia sets appear quasiconformally in the Mandelbrot set

Tomoki Kawahira, Masashi Kisaka

Abstract

In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". We can also find images of a Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. All the parameters belonging to these images are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set.

Julia sets appear quasiconformally in the Mandelbrot set

Abstract

In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a parabolic or Misiurewicz Julia set. Furthermore, zoom in its middle part, then we can see a certain nested structure ("decoration") and finally another "smaller Mandelbrot set" appears. A similar nested structure exists in the Julia set for any parameter in the "smaller Mandelbrot set". We can also find images of a Julia sets by quasiconformal maps with dilatation arbitrarily close to 1. This answers a question by Adrian Douady. All the parameters belonging to these images are semihyperbolic and this leads to the fact that the set of semihyperbolic but non-Misiurewicz and non-hyperbolic parameters is dense with Hausdorff dimension 2 in the boundary of the Mandelbrot set.

Paper Structure

This paper contains 47 sections, 15 theorems, 122 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The Model Sets and the Statements of the Results
  3. The quadratic-like maps and the Mandelbrot-like family
  4. Proof of Theorem A for the Misiurewicz case
  5. Proof.
  6. Proof.
  7. Claim.
  8. Proof.
  9. Proof.
  10. Proof of Theorem A for the parabolic case
  11. Proof of Theorem B
  12. Proof.
  13. Almost conformal straightenings
  14. Proof.
  15. Proof.
  16. ...and 32 more sections

Key Result

Theorem 1

Let then the Hausdorff dimension of $SH$ is 2. In particular, the Hausdorff dimension of the boundary of $M$ is 2.

Figures (11)

  • Figure 1: Zooms around a Misiurewicz point $c_1 = s_0 \perp c_0$ in a primitive small Mandelbrot set $M_{s_0}$, where $c_0$ is a Misiurewicz parameter satisfying $P_{c_0}(P_{c_0}^4(0)) = P_{c_0}^4(0)$. After a sequence of nested structures, another smaller Mandelbrot set $M_{s_1}$ appears in (15). Here, $s_0 \approx 0.3591071125276155 + 0.6423830938166145i$, $c_0 \approx -0.1010963638456221 + 0.9562865108091415i$, $c_1 \approx 0.3626697754647427 + 0.6450273437137847i$ and $s_1 \approx 0.3626684938191616 + 0.6450238859863952i$. The widths of the figures (1) and (15) are about $10^{-1.5}$ and $10^{-11.9}$, respectively.
  • Figure 2: Zooms around the critical point $0$ in $K(P_{s_1 \perp c})$ for $s_1 \perp c$ in $M_{s_1}$, which is the smaller Mandelbrot set in Figure \ref{['figures of a primitive-Misiurewicz case']}--(15) and $c \in M$ is the parameter for the Douady rabbit. $s_1 \approx 0.3626684938191616+0.6450238859863952i$, $c \approx -0.12256+0.74486i$ and $s_1 \perp c \approx 0.3626684938192285 + 0.6450238859865394i$.
  • Figure 3: The first row depicts the decorated Mandelbrot set $\mathcal{M}(c')$ for $c'= -0.10 + 0.97i$ (close to the Misiurewicz parameter $c_0 \approx -0.1011+0.9563i$, the landing point of the external ray of angle $11/56$) and $R=220$. The second row depicts the set $\bigcup_{m \ge 0} \Gamma_m(c')$. The third row depicts the decorated filled Julia set $\mathcal{K}_c(c')$ for $c \approx -0.123+0.745$ (the rabbit).
  • Figure 4: The original Mandelbrot set (left), a "satellite" small Mandelbrot set (middle), and a "primitive" small Mandelbrot set (right). The stars indicate the central superattracting parameters.
  • Figure 5: (i): The decorated Mandelbrot set $\mathcal{M}(c')$ for $c'=-0.77+0.18 i$ (close to the parabolic parameter $c_0=-0.75$. (ii) and (iii): Embedded quasiconformal copies of $\mathcal{M}(c')$ above near the satellite/primitive small Mandelbrot sets in Figure \ref{['primitive and satellite small M-set']}.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2: Semihyperbolicity
  • Theorem : Shishikura, 1998
  • Remark
  • Proposition 3.1
  • Lemma 4.1
  • Remark
  • Lemma 4.2
  • Lemma 4.3
  • Remark
  • ...and 11 more