The dynamics of holomorphic correspondences of P^1: invariant measures and the normality set

Abstract

This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if $F$ is a holomorphic correspondence on $\mathbb{P}^1$, then (under certain conditions) $F$ admits a measure $\mu_F$ such that, for any point $z$ drawn from a "large" open subset of $\mathbb{P}^1$, $\mu_F$ is the weak*-limit of the normalised sums of point masses carried by the pre-images of $z$ under the iterates of $F$. Let ${}^\dagger{F}$ denote the transpose of $F$. Under the condition $d_{top}(F) > d_{top}({}^\dagger{F})$, where $d_{top}$ denotes the topological degree, the above phenomemon was established by Dinh and Sibony. We show that the support of this $\mu_F$ is disjoint from the normality set of $F$. There are many interesting correspondences on $\mathbb{P}^1$ for which $d_{top}(F) \leq d_{top}({}^\dagger{F})$. Examples are the correspondences introduced by Bullett and collaborators. When $d_{top}(F) \leq d_{top}({}^\dagger{F})$, equidistribution cannot be expected to the full extent of Brolin's theorem. However, we prove that when $F$ admits a repeller, equidistribution in the above sense holds true.