Baker Omitted Value

TL;DR

This paper introduces the concept of Baker omitted value (bov) for entire and meromorphic functions, linking it to properties like Baker wandering domains and invariant Fatou components in complex dynamics.

Contribution

It defines the Baker omitted value and establishes its equivalence with the unbounded image of unbounded curves, also relating it to the existence of Baker wandering domains.

Findings

Entire functions with bov have unbounded images of unbounded curves.

Functions with bov can have at most one completely invariant Fatou component.

Presence of bov implies the existence of Baker wandering domains.

Abstract

We define Baker omitted value, in short bov, of an entire or meromorphic function f in the complex plane as an omitted value for which there exists r0 > 0 such that for each ball Dr(a) centered at a and with radius r satisfying 0 < r < r0, every component of the boundary of f only asymptotic value. An entire function has bov if and only if the image of every unbounded curve is unbounded. It follows that an entire function has bov whenever it has a Baker wandering domain. Functions with bov has at most one completely invariant Fatou component.