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){∞ } is always connected and that, if I^\!+\!(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded.