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

Published online by Cambridge University Press:  23 March 2016

J. W. OSBORNE ,
P. J. RIPPON  and
G. M. STALLARD
J. W. OSBORNE
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK email john.osborne@open.ac.uk, phil.rippon@open.ac.uk, gwyneth.stallard@open.ac.uk
P. J. RIPPON
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK email john.osborne@open.ac.uk, phil.rippon@open.ac.uk, gwyneth.stallard@open.ac.uk
G. M. STALLARD
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK email john.osborne@open.ac.uk, phil.rippon@open.ac.uk, gwyneth.stallard@open.ac.uk
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Abstract

We investigate the connectedness properties of the set $I^{\!+\!}(f)$ of points where the iterates of an entire function $f$ are unbounded. In particular, we show that $I^{\!+\!}(f)$ is connected whenever iterates of the minimum modulus of $f$ tend to $\infty$. For a general transcendental entire function $f$, we show that $I^{\!+\!}(f)\cup \{\infty \}$ is always connected and that, if $I^{\!+\!}(f)$ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 4 , June 2017 , pp. 1291 - 1307
Copyright
© Cambridge University Press, 2016 

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