The dynamics of holomorphic correspondences of P1: invariant measures and the normality set

This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if F is a holomorphic correspondence on 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 P-1, mu F is the weak*- limit of the normalized 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 dtop(F) > d(top)((+) F), where dtop denotes the topological degree, the above phenomenon 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 P-1 for which d(top)(F) <= d(top)((+) F). Examples are the correspondences introduced by Bullett and collaborators. When d(top)(F) = 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.