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

TL;DR

This paper provides simple proofs for derivative estimates of holomorphic motions of Julia sets near two boundary points of the Mandelbrot set, demonstrating how Julia sets converge in Hausdorff distance as parameters approach these points.

Contribution

The paper introduces concise proofs for derivative bounds of Julia set motions near boundary points 1/4 and -2, establishing precise convergence rates.

Findings

Derivative of Julia set points is O(1/√(1/4 - c)) near c=1/4

Derivative of Julia set points is O(1/√(-2 - c)) near c=-2

Hausdorff distance between Julia sets scales as √(1/4 - c) near c=1/4

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 $\sqrt{1/4-c}$.