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

TL;DR

This paper studies the connectedness of the set where an entire function's iterates are unbounded, showing conditions under which this set is connected or has uncountably many components.

Contribution

It establishes new connectedness results for the escaping set of entire functions, linking it to the behavior of the minimum modulus and providing a comprehensive topological analysis.

Findings

$ I^{+}(f) $ is connected if iterates of the minimum modulus tend to infinity.

$ I^{+}(f)  ty $ is always connected when unioned with infinity.

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