On Fatou Sets Containing Baker Omitted Value

Abstract

An omitted value of a transcendental meromorphic function f is called a Baker omitted value, in short bov if there is a disk D centered at the bov such that each component of the boundary of f −1 (D) is bounded. Assuming the bov to be in the Fatou set, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If U is the Fatou component containing the bov then it is proved that a Fatou component U is infinitely connected if and only if it lands on U , i.e. f k (U ) ⊂ U for some k ≥ 1. Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbit does not intersect U is finite, we have shown that the connectivity of each Fatou component belongs to a finite set independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent. It is also proved that, if there is a 2-periodic Baker domain, or a 2-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components.

Neurotrauma at the All India Institute of Medical Sciences Bhubaneswar: An Overview

“…This is probably the second instance of examples of entire maps with bov, the first being the function given by Baker in 1975 as an infinite product [1]. The first result of this article is a generalization of Remark 2.2, [8].…”

“…Recently, it is proved that e z + cz d has bov for every non-zero complex number c and every natural number d (see Remark 2.2, [8]). This is probably the second instance of examples of entire maps with bov, the first being the function given by Baker in 1975 as an infinite product [1].…”

“…For an entire function, the bov (that is always ∞) is always a limit point of critical values (see Lemma 2.5, [8] for a proof for meromorphic functions which also works for entire functions). Therefore the set of (infinitely many) critical values of E k (z) + P (z) is unbounded.…”

“…It is important to note that the bov of a meromorphic function is always a finite complex number. Lemma 2.5 of [8] states that if a meromorphic function has a bov then the bov is a limit point of its critical values. This gives rise to a question: when the bov is the only limit point of critical values?…”