Connectedness properties of the set where the iterates of an entire function are unbounded. (23rd March 2016) - OSBORNE, J. W.; RIPPON, P. J.; STALLARD, G. M.
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 $I^{\!+\!}(f)\cup \{∞ \}$ is always connected and that, if $I^{\!+\!}(f)$ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.