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Transactions of the American Mathematical Society

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From Cantor to semi-hyperbolic parameters along external rays


Authors: Yi-Chiuan Chen and Tomoki Kawahira
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37F45; Secondary 37F99
DOI: https://doi.org/10.1090/tran/7839
Published electronically: June 17, 2019
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Abstract: For the quadratic family $ f_{c}(z) = z^2+c$ with $ c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $ \hat {c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $ z = z(c)$ in the Julia set, the derivative $ dz(c)/dc$ is uniformly $ O(1/\sqrt {\vert c-\hat {c}\vert})$ when $ c$ belongs to a parameter ray that lands on $ \hat {c}$. We also characterize the degeneration of the dynamics along the parameter ray.


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Additional Information

Yi-Chiuan Chen
Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
Email: YCChen@math.sinica.edu.tw

Tomoki Kawahira
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan; and Mathematical Science Team, RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
Email: kawahira@math.titech.ac.jp

DOI: https://doi.org/10.1090/tran/7839
Received by editor(s): March 8, 2018
Received by editor(s) in revised form: December 24, 2018, and March 2, 2019
Published electronically: June 17, 2019
Additional Notes: The first author was partly supported by NSC 99-2115-M-001-007, MOST 103-2115-M-001-009, 104-2115-M-001-007, and 105-2115-M-001-003
The second author was partly supported by JSPS KAKENHI Grant Number 16K05193
Article copyright: © Copyright 2019 American Mathematical Society