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

Osborne, John W. and Sixsmith, David J. (2016). On the set where the iterates of an entire function are neither escaping nor bounded. Annales Academiae Scientiarum Fennicae Mathematica, 41 pp. 561–578.

DOI: https://doi.org/10.5186/aasfm.2016.4134

Abstract

For a transcendental entire function ƒ, we study the set of points BU(ƒ) whose iterates under ƒ 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(ƒ), then most boundary points of U (in the sense of harmonic measure) lie in BU(ƒ). We prove this using a new result concerning the set of limit points of the iterates of ƒ on the boundary of a wandering domain. Finally, we give some examples to illustrate our results.

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