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

Osborne, John; Rippon, Philip and Stallard, Gwyneth (2017). Connectedness properties of the set where the iterates of an entire function are unbounded. Ergodic Theory and Dynamical Systems, 37(4) pp. 1291–1307.

DOI: https://doi.org/10.1017/etds.2015.85

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 ∞. For a general transcendental entire function f, we show that I+(f)∪ 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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