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Let $f$ be a transcendental entire function and let $I(f)$ denote the set ofpoints that escape to infinity under iteration. We give conditions which ensurethat, 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 hasorder 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 alsogive a new criterion related to $I(f)$ which is sufficient to ensure that $f$has no unbounded Fatou components.

Escaping points of entire functions of small growth