  We highlight a condition, the approaching geodesics property, on a proper
geodesic Gromov hyperbolic metric space, which implies that the horofunction
compactification is topologically equivalent to the Gromov compactification. It
is known that this equivalence does not hold in general. We prove using
rescaling techniques that the approaching geodesics property is satisfied by
bounded strongly pseudoconvex domains of $\mathbb{C}^q$ endowed with the
Kobayashi metric. We also show that bounded convex domains of $\mathbb{C}^q$
with boundary of finite type in the sense of D'Angelo satisfy a weaker
property, which still implies the equivalence of the two said
compactifications. As a consequence we prove that on those domains big and
small horospheres as defined by Abate coincide. Finally we generalize the
classical Julia's lemma, giving applications to the dynamics of non-expanding
maps.
