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

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

17 pages, 2 figures

References in corpus (2)

• On a question of Eremenko concerning escaping components of entire functions
• On the set where the iterates of an entire function are neither escaping nor bounded

Cited by in corpus (1)

• Dynamical sets whose union with infinity is connected