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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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From Cantor to semi-hyperbolic parameters along external rays
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by Yi-Chiuan Chen and Tomoki Kawahira PDF
Trans. Amer. Math. Soc. 372 (2019), 7959-7992 Request permission

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 $O(1/\sqrt {|c-\hat {c}|})$ 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
  • MR Author ID: 725580
  • 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
  • MR Author ID: 661650
  • Email: kawahira@math.titech.ac.jp
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7959-7992
  • MSC (2010): Primary 37F45; Secondary 37F99
  • DOI: https://doi.org/10.1090/tran/7839
  • MathSciNet review: 4029687