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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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Julia sets as buried Julia components
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by Youming Wang and Fei Yang PDF
Trans. Amer. Math. Soc. 373 (2020), 7287-7326 Request permission

Abstract:

Let $f$ be a rational map with degree $d\geq 2$ whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map $g$ such that $g$ contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of $f$ on the Julia set if and only if $f$ does not have parabolic basins and Siegel disks. If such $g$ exists, then the degree can be chosen such that $\deg (g)\leq 7d-2$. In particular, if $f$ is a polynomial, then $g$ can be chosen such that $\deg (g)\leq 4d+4$. Moreover, some quartic and cubic rational maps whose Julia sets contain buried Jordan curves are also constructed.
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Additional Information
  • Youming Wang
  • Affiliation: Department of Applied Mathematics, Hunan Agricultural University, Changsha, 410128, People’s Republic of China
  • Email: wangym2002@163.com
  • Fei Yang
  • Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
  • MR Author ID: 983714
  • Email: yangfei@nju.edu.cn
  • Received by editor(s): December 14, 2017
  • Received by editor(s) in revised form: October 19, 2019, February 22, 2020, and February 26, 2020
  • Published electronically: July 29, 2020
  • Additional Notes: This work was supported by the National Natural Science Foundation of China (grant No. 11671092), the Natural Science Foundation of Jiangsu Province (No. BK20191246), the Fundamental Research Funds for the Central Universities (grant No. 0203-14380025), and EDF of Hunan Province (grant No. 16C0763).
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 7287-7326
  • MSC (2010): Primary 37F45; Secondary 37F10
  • DOI: https://doi.org/10.1090/tran/8144
  • MathSciNet review: 4155208