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arXiv · 1504.05611

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

J. W. Osborne·P. J. Rippon·G. M. Stallard
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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 infinity. For a general transcendental entire function $ f $, we show that $ I^{\!+\!}(f) \cup \lbrace \infty \rbrace $ is always connected and that, if $ I^{\!+\!}(f) $ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.

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J. W. Osborne, P. J. Rippon, G. M. Stallard. 2015-04-21. Connectedness properties of the set where the iterates of an entire function are unbounded. https://doi.org/10.1017/etds.2015.85

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