Connectedness properties of the set where the iterates of an entire function are bounded

TL;DR

This paper explores the connectedness of the set of points where an entire function's iterates are bounded, establishing new conditions under which this set's structure can be characterized, including examples and constructions.

Contribution

It introduces a class of transcendental entire functions where an analogue of the Branner-Hubbard conjecture applies and analyzes the topological complexity of their boundedness sets.

Findings

If K(f) is disconnected, it has uncountably many components.

K(f) can be totally disconnected.

Constructed examples show K(f) can have a component with empty interior that is not a singleton.

Abstract

We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. In particular, we describe a class of transcendental entire functions for which an analogue of the Branner-Hubbard conjecture holds and show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples to show that K(f) can be totally disconnected, and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.