  In this paper, we present a new technique for studying the dynamics of a
finitely generated rational semigroup. Such a semigroup can be associated
naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then,
results on the iterative dynamics of such a correspondence can be applied to
the study of the rational semigroup. We focus on a certain invariant measure
for the aforementioned correspondence---known as the equilibrium measure. This
confers some advantages over many of the known techniques for studying the
dynamics of rational semigroups. We use the equilibrium measure to analyse the
distribution of repelling fixed points of a finitely generated rational
semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia
set of such a semigroup.
