Dynamical sets whose union with infinity is connected

Abstract

Suppose that $f$ is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider's web. We use our results to give a large class of functions in the Eremenko-Lyubich class for which the escaping set is not a spider's web. Finally we give a novel topological criterion for certain sets to be a spider's web.

References in corpus (5)

• Hyperbolic dimension and radial Julia sets of transcendental functions
• On the set where the iterates of an entire function are neither escaping nor bounded
• A landing theorem for entire functions with bounded post-singular sets
• Non-escaping endpoints do not explode
• Connectedness properties of the set where the iterates of an entire function are unbounded

Cited by in corpus (4)

• Orbifold expansion and entire functions with bounded Fatou components
• The bungee set in quasiregular dynamics
• The escaping set in transcendental dynamics
• Spiders' webs in the Eremenko-Lyubich class