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

TL;DR

This paper investigates the behavior of invariant measures for holomorphic correspondences on the Riemann sphere, establishing conditions under which equidistribution occurs and analyzing the relationship between the support of these measures and the normality set.

Contribution

It extends the understanding of invariant measures for holomorphic correspondences, especially in cases where the topological degrees are equal or the correspondence admits a repeller.

Findings

Invariant measure $$ exists under certain conditions.

Support of $$ is disjoint from the normality set.

Equidistribution holds when the correspondence admits a repeller.

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.