Holomorphic correspondences related to finitely generated rational semigroups

TL;DR

This paper introduces a novel approach using holomorphic correspondences to analyze the dynamics of finitely generated rational semigroups, providing new insights into their invariant measures and Julia set dimensions.

Contribution

It develops a new technique linking rational semigroup dynamics to holomorphic correspondences, enabling analysis via equilibrium measures.

Findings

Distribution of repelling fixed points characterized

Sharp bound for Hausdorff dimension of Julia set established

Advantages over existing methods demonstrated

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 $\mathbb{P}^1$. 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.