Rational maps without Herman rings

TL;DR

This paper proves that rational maps with degree at least two having Herman rings must have at least two disjoint infinite critical orbits in their Julia set, establishing a sharp criterion for the existence of Herman rings.

Contribution

It establishes a new criterion linking the number of infinite critical orbits in the Julia set to the existence of Herman rings in rational maps.

Findings

Rational maps with Herman rings have at least two disjoint infinite critical orbits.

Existence of a cubic rational map with exactly two critical orbits and a Herman ring.

Rational maps with at most one infinite critical orbit in the Julia set have no 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.