Rational maps without Herman rings

Yang, Fei

doi:10.1090/proc/13336

Rational maps without Herman rings
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Proc. Amer. Math. Soc. 145 (2017), 1649-1659 Request permission

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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