  There are several classes of transcendental entire functions for which the
Julia set consists of an uncountable union of disjoint curves each of which
joins a finite endpoint to infinity. Many authors have studied the topological
properties of this set of finite endpoints. It was recently shown that, for
certain functions in the exponential family, there is a strong dichotomy
between the topological properties of the set of endpoints which escape and
those of the set of endpoints which do not escape. In this paper, we show that
this result holds for large families of functions in the Eremenko-Lyubich
class. We also show that this dichotomy holds for a family of functions,
outside that class, which includes the much-studied Fatou function defined by
$f(z) := z + 1+ e^{-z}.$ Finally, we show how our results can be used to
demonstrate that various sets are spiders' webs, generalising results such as
those in a recent paper of the first author.
