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

Osborne, John (2013). Connectedness properties of the set where the iterates of an entire function are bounded. Mathematical Proceedings of the Cambridge Philosophical Society, 155(03) pp. 391–410.

DOI: https://doi.org/10.1017/S0305004113000455

Abstract

We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K(f) is totally disconnected if and only if each component of K(f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is 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.