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 ${}^{†}F$ denote the transpose of $F$. Under the condition $d_{top}(F)>d_{top}({}^{†}F)$, 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 $d_{top}(F)\le d_{top}({}^{†}F)$. Examples are the correspondences introduced by Bullett and collaborators. When $d_{top}(F)\le d_{top}({}^{†}F)$, 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.

24 pages; Section 3 significantly shortened, typos in the proof of Theorem 3.2 removed and Remark 5.3 added; has appeared in Complex Var. Elliptic Equ. as referenced below

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