  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.
