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No Herman rings for regularly ramified rational maps

Authors: Jun Hu and Yingqing Xiao
Journal: Proc. Amer. Math. Soc. 147 (2019), 1587-1596
MSC (2010): Primary 37F10, 37F45; Secondary 30F40
Published electronically: January 9, 2019
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Abstract: It is due to Milnor that there is no Herman ring for any rational map with two critical values. In this paper, we are concerned with if there exist Herman rings for rational maps with three critical values. We first prove that such rational maps cannot have Herman rings of period $ 1$, $ 2$, or $ 3$. Then we prove all regularly ramified rational maps have no Herman rings in their Fatou sets.

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Additional Information

Jun Hu
Affiliation: Department of Mathematics, Brooklyn College of CUNY, Brooklyn, New York 11210 –and– Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, New York 10016 –and– NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, People’s Republic of China

Yingqing Xiao
Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha 410082, People’s Republic of China

Keywords: Fatou set, Herman ring, regularly ramified rational map
Received by editor(s): March 24, 2018
Received by editor(s) in revised form: July 15, 2018, and July 22, 2018
Published electronically: January 9, 2019
Additional Notes: The research of the first author was supported by PSC-CUNY research awards and a visiting professorship at NYU Shanghai.
The research of the second author was supported by the National Natural Science Foundation of China under grant numbers 11301165, 11371126, and 11571099.
The second author was the corresponding author.
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society