Rational maps without Herman rings

Abstract

Let $f$ be a rational map with degree at least two. We prove that $f$ has at least $2$ 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.

10 pages, 3 figures, to appear in Proc. Amer. Math. Soc

References in corpus (1)

• On the dynamics of a family of renormalization transformations

Cited by in corpus (1)

• On the dynamics of a family of renormalization transformations