Holomorphic correspondences related to finitely generated rational semigroups

Abstract

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 [Formula: see text]. Results on the iterative dynamics of such a correspondence can now be applied to the study of the rational semigroup. We focus on an 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 analyze 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.