  In recent years, there has been significant progress in the understanding of
the dynamics of transcendental entire functions with bounded postsingular set.
In particular, for certain classes of such functions, a complete description of
their topological dynamics in terms of a simpler model has been given inspired
by methods from polynomial dynamics. In this paper, and for the first time, we
give analogous results in cases when the postsingular set is unbounded. More
specifically, we show that if $f$ is of finite order, has bounded criticality
on its Julia set $J(f)$, and its singular set consists of finitely many
critical values that escape to infinity and satisfy a certain separation
condition, then $J(f)$ is a collection of dynamic rays or hairs, that split at
critical points, together with their corresponding landing points. In fact, our
result holds for a much larger class of functions with bounded singular set.
Moreover, this result is a consequence of a significantly more general one: we
provide a topological model for the action of $f$ on its Julia set.
