%0 Journal Article
%T Escaping points of entire functions of small growth
%A P. J. Rippon
%A G. M. Stallard
%J Mathematics
%D 2008
%I arXiv
%X Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.
%U http://arxiv.org/abs/0801.3605v1