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Julia sets appear quasiconformally in the Mandelbrot set, II: A parabolic proof

Tomoki Kawahira, Masashi Kisaka

TL;DR

The paper proves that Julia sets appear quasiconformally in the Mandelbrot set near any boundary point $c_0\in\partial M$ by a parabolic proof. It replaces the previous Misiurewicz-based approach with parabolic implosion, constructing two nested quadratic-like families and a tubing to transfer decorated copies of the Mandelbrot set into parameter space. The key contribution is an explicit, alternative parabolic construction of decorated Mandelbrot copies $\mathcal{M}(\sigma)$ whose associated Julia sets $J(P_{\sigma})$ appear inside $M$ in a controlled, quasiconformal way. This advances understanding of the fine-scale self-similarity at $\partial M$ and shows that Cantor Julia sets can be realized inside $M$ in a neighborhood of any boundary point.

Abstract

Following the ideas of A.~Douady, we give an alternative proof of the authors' result: for any boundary point $c_0$ of the Mandelbrot set $M$, we can find small quasiconformal copies of $M$ in $M$ that are encaged in nested quasiconformal copies of the totally disconnected Julia set of a parameter arbitrarily close to $c_0$.

Julia sets appear quasiconformally in the Mandelbrot set, II: A parabolic proof

TL;DR

The paper proves that Julia sets appear quasiconformally in the Mandelbrot set near any boundary point by a parabolic proof. It replaces the previous Misiurewicz-based approach with parabolic implosion, constructing two nested quadratic-like families and a tubing to transfer decorated copies of the Mandelbrot set into parameter space. The key contribution is an explicit, alternative parabolic construction of decorated Mandelbrot copies whose associated Julia sets appear inside in a controlled, quasiconformal way. This advances understanding of the fine-scale self-similarity at and shows that Cantor Julia sets can be realized inside in a neighborhood of any boundary point.

Abstract

Following the ideas of A.~Douady, we give an alternative proof of the authors' result: for any boundary point of the Mandelbrot set , we can find small quasiconformal copies of in that are encaged in nested quasiconformal copies of the totally disconnected Julia set of a parameter arbitrarily close to .

Paper Structure

This paper contains 7 sections, 5 theorems, 64 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quadratic-like maps and renormalization
  3. Fatou coordinates
  4. Step (P1): Definitions of $U_c$, $U_c'$ and $V_c$
  5. Step (P2): Construction of the quadratic-like family $\boldsymbol{G}$
  6. Step (P3): Proof for $\boldsymbol{G}$ being a Mandelbrot-like family
  7. Step (P4): Existence of a copy of ${\mathcal{M}}(c_0+\eta)$ in $W$

Key Result

Theorem 1.1

For any choices of $c_0 \in \partial M$ and $\varepsilon>0$, there exists a parameter such that $M$ contains a quasiconformal copy of the decorated Mandelbrot set ${\mathcal{M}}(\sigma)$. Moreover, one can find such a copy in any open disk intersecting with $\partial M$.

Figures (5)

  • Figure 1: (i): The decorated Mandelbrot set $\mathcal{M}(\sigma)$ for $\sigma=-0.77+0.18 i$ (close to the parabolic parameter $c_0=-0.75$). (ii) and (iii): Embedded quasiconformal copies of $\mathcal{M}(\sigma)$ above near satellite and primitive small Mandelbrot sets.
  • Figure 2: We choose a pair of repelling and attracting petals. Their intersection has two components when $\nu=1$.
  • Figure 3: The sector $S$ for $\nu=1$ (left) and $\nu \ge 2$ (right).
  • Figure 4: A typical behavior of the critical orbit by $f_c^{k \nu }$ near $q_c$ for $\nu =3$.
  • Figure 5: The Jordan domains $U_c$, $U_c'$, $V_c$, and $V'_c$ for parameters $c \in S \smallsetminus \{c_1\}$. When $c = c_1$, the domains $U_{c_1}$, $U_{c_1}'$, and $V_{c_1}$ are arranged as in the figure, but the small Julia set $J(f_{c_1})$ is connected. The additional domain $V_c'$ will be defined later and exists only for $c \in S \smallsetminus \{c_1\}$.

Theorems & Definitions (5)

  • Theorem 1.1: Julia sets appear quasiconformally
  • Theorem 1.2: Theorem A of Kawahira-Kisaka 2018
  • Lemma 4.1: cf. Lemma 4.1 of Kawahira-Kisaka 2018
  • Lemma 5.1: cf. Lemma 4.2 of Kawahira-Kisaka 2018
  • Lemma 6.1