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We investigate the connectedness properties of the set $ I^{\!+\!}(f) $ ofpoints where the iterates of an entire function $ f $ are unbounded. Inparticular, we show that $ I^{\!+\!}(f) $ is connected whenever iterates of theminimum modulus of $ f $ tend to infinity. For a general transcendental entirefunction $ f $, we show that $ I^{\!+\!}(f) \cup \lbrace \infty \rbrace $ isalways connected and that, if $ I^{\!+\!}(f) $ is disconnected, then it hasuncountably many components, infinitely many of which are unbounded.

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