Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 561-578

ON THE SET WHERE THE ITERATES OF AN ENTIRE FUNCTION ARE NEITHER ESCAPING NOR BOUNDED

John W. Osborne and David J. Sixsmith

The Open University, Department of Mathematics and Statistics
Walton Hall, Milton Keynes MK7 6AA, United Kingdom; john.osborne 'at' open.ac.uk

The Open University, Department of Mathematics and Statistics
Walton Hall, Milton Keynes MK7 6AA, United Kingdom; david.sixsmith 'at' open.ac.uk

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.

2010 Mathematics Subject Classification: Primary 37F10; Secondary 30D05.

Key words: Complex dynamics, entire functions, non-escaping unbounded orbits, wandering domains, harmonic measure.

Reference to this article: J. W. Osborne and D. J. Sixsmith: On the set where the iterates of an entire function are neither escaping nor bounded. Ann. Acad. Sci. Fenn. Math. 41 (2016), 561-578.

Full document as PDF file

doi:10.5186/aasfm.2016.4134