Simple proofs for the derivative estimates of the holomorphic motion near two boundary points of the Mandelbrot set

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:473

Article

Chen, Yi-Chiuan¹  Kawahira, Tomoki^2,3 

[1] Acad Sinica, Inst Math, Taipei 10617, Taiwan

[2] Tokyo Inst Technol, Dept Math, Tokyo 1528551, Japan

[3] RIKEN, Math Sci Team, Ctr Adv Intelligence Project AIP, Chuo Ku, 1-4-1 Nihonbashi, Tokyo 1030027, Japan

关键词: Quadratic map;    Holomorphic motion;    Hausdorff convergence;   

DOI  :  10.1016/j.jmaa.2018.12.052

来源: Elsevier

【 摘 要 】

For the complex quadratic family q(c) : z bar right arrow 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/root 1/4 - c) when real c NE arrow 1/4; and is uniformly O(1/root-2 - c) when real c NE arrow -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 <= c < 1/4 and J(q(1/4)) is exactly root 1/4 - c. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free