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Rational maps without Herman rings


Author: Fei Yang
Journal: Proc. Amer. Math. Soc. 145 (2017), 1649-1659
MSC (2010): Primary 37F45; Secondary 37F10, 37F30
DOI: https://doi.org/10.1090/proc/13336
Published electronically: October 13, 2016
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Abstract: Let $ f$ be a rational map with degree at least two. We prove that $ f$ has at least two disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic rational map having exactly two critical grand orbits but also having a Herman ring. In particular, $ f$ has no Herman rings if it has at most one infinite critical orbit in the Julia set. These criterions derive some known results about the rational maps without Herman rings.


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Additional Information

Fei Yang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China
Email: yangfei@nju.edu.cn

DOI: https://doi.org/10.1090/proc/13336
Keywords: Herman rings, quasiconformal surgery, Julia set, Fatou set
Received by editor(s): March 17, 2016
Received by editor(s) in revised form: June 12, 2016
Published electronically: October 13, 2016
Communicated by: Nimish Shah
Article copyright: © Copyright 2016 American Mathematical Society