On the set where the iterates of an entire function are neither escaping nor bounded

TL;DR

This paper investigates the set of points for transcendental entire functions whose iterates are neither escaping nor bounded, revealing new connectedness properties and boundary behaviors of these points.

Contribution

It introduces new results on the connectedness of BU(f) and boundary point distribution in Fatou components for transcendental entire functions.

Findings

Most boundary points of Fatou components meeting BU(f) lie in BU(f).

Connectedness properties of BU(f) are characterized.

Examples illustrate the theoretical results.

Abstract

For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this set and show that, if U is a Fatou component that meets BU(f), then most boundary points of U (in the sense of harmonic measure) lie in BU(f). We prove this using a new result concerning the set of limit points of the iterates of f on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.