Rotation domains and Stable Baker omitted value

TL;DR

This paper investigates the dynamics of transcendental meromorphic functions with Baker omitted values, establishing finiteness of certain Herman rings, conditions for Julia components to be singletons, and the behavior of boundaries of Fatou and rotation domains.

Contribution

It provides new results on the structure of Herman rings, Julia components, and Fatou domain boundaries in functions with Baker omitted values, under various critical point assumptions.

Findings

Number of Herman rings of a given period is finite.

Julia components intersect finitely many Herman rings.

Boundaries of wandering domains and rotation domains are in the omega-limit set of recurrent critical points.

Abstract

A Baker omitted value, in short \textit{bov} of a transcendental meromorphic function $f$ is an omitted value such that there is a disk $D$ centered at the bov for which each component of the boundary of $f^{-1}(D)$ is bounded. Assuming all the iterates $f^n$ are analytic in a neighborhood of its bov, this article proves that the number of Herman rings of a particular period is finite and every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that $f$ has infinitely many repelling fixed points. If a repelling periodic point of period $p$ is on the boundary of a $p$-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be singleton. As a consequence, it is proved that if the boundary of a wandering domain $W$ accumulates at some point of the plane under the iteration of $f$ then each limit of $f^n$ on $W$ is either a parabolic periodic point or in the $\omega$-limit set of recurrent critical points. Using the same ideas, the boundary of rotation domains are shown to be in the $\omega$-limit set of recurrent critical points.