Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set

Yi-Chiuan Chen, Tomoki Kawahira

Abstract

For the complex quadratic family $q_c:z\mapsto z^2+c$, it is known that every point in the Julia set $J(q_c)$ moves holomorphically on $c$ except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the following derivative estimates of the motions near the boundary points $1/4$ and $-2$: for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt{1/4-c})$ when real $c\nearrow 1/4$; and is uniformly $O(1/\sqrt{-2-c})$ when real $c\nearrow -2$. These estimates of the derivative imply Hausdorff convergence of the Julia set $J(q_c)$ when $c$ approaches these boundary points. In particular, the Hausdorff distance between $J(q_c)$ with $0\le c<1/4$ and $J(q_{1/4})$ is exactly $\sqrt{1/4-c}$.

Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set

Abstract

For the complex quadratic family , it is known that every point in the Julia set moves holomorphically on except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the following derivative estimates of the motions near the boundary points and : for each in the Julia set, the derivative is uniformly when real ; and is uniformly when real . These estimates of the derivative imply Hausdorff convergence of the Julia set when approaches these boundary points. In particular, the Hausdorff distance between with and is exactly .

Paper Structure

This paper contains 13 sections, 7 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hausdorff convergence and dynamical degeneration
  3. $c\nearrow 1/4$ or $\mu\searrow 1$
  4. Hausdorff distance.
  5. Notation.
  6. Proof.
  7. $c\nearrow -2$ or $\mu\searrow 4$
  8. Proof of Theorem \ref{['Main_thm_2']}
  9. Proof.
  10. Proof of Theorem \ref{['Main_thm']}
  11. Proof.
  12. Non-pre-fixed case.
  13. Pre-fixed case.

Key Result

Theorem 1.1

Let $\hat{c} \in \partial \mathbb{M}$ be a semi-hyperbolic parameter that is the landing point of $\mathcal{R}(\theta)$. Then there exists a constant $K>0$ that depends only on $\hat{c}$ such that for any $c\in\mathcal{R}(\theta)$ sufficiently close to ${\hat{c}}$ and any $z = z(c)\in J(q_c)$, the p

Figures (2)

  • Figure 1: Top: The Julia set $J(q_c)$ for $c= k/20~ (k=0, 1, \cdots, 5)$. Bottom: Real analytic motion of the preimages of the repelling fixed point for $0 \le c <1/4$.
  • Figure 2: Real analytic motion of the Julia set $J(f_\mu)$ for $\mu \searrow 4$. See Figure 2 of CK for the corresponding motion for $q_c$ for $c \nearrow -2$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 3.1
  • ...and 2 more