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Transactions of the American Mathematical Society

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Julia sets as buried Julia components

Authors: Youming Wang and Fei Yang
Journal: Trans. Amer. Math. Soc. 373 (2020), 7287-7326
MSC (2010): Primary 37F45; Secondary 37F10
Published electronically: July 29, 2020
MathSciNet review: 4155208
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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

Fei Yang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
MR Author ID: 983714

Keywords: Julia sets, buried component, singular perturbations
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).
Article copyright: © Copyright 2020 American Mathematical Society