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

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.

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• Dynamical sets whose union with infinity is connected
• Connectedness properties of the set where the iterates of an entire function are unbounded
• Eremenko's conjecture, wandering Lakes of Wada, and maverick points
• The escaping set in transcendental dynamics
• Oscillating Wandering Domains for Functions with Escaping Singular Values
• On the dimension of points which escape to infinity at given rate under exponential iteration
• On slow escaping and non-escaping points of quasimeromorphic mappings