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

Published online by Cambridge University Press:  04 July 2013

JOHN OSBORNE
JOHN OSBORNE*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA. e-mail: [email protected]
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 391 - 410
Copyright
Copyright © Cambridge Philosophical Society 2013 

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