Rational maps without Herman rings
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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.References
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Additional Information
- 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): March 17, 2016
- Received by editor(s) in revised form: June 12, 2016
- Published electronically: October 13, 2016
- Communicated by: Nimish Shah
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1649-1659
- MSC (2010): Primary 37F45; Secondary 37F10, 37F30
- DOI: https://doi.org/10.1090/proc/13336
- MathSciNet review: 3601556