Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets

In this paper, we analyze a certain family of holomorphic correspondences on $\widehat{\mathbb{C}}\times \widehat{\mathbb{C}}$ and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map $F$ is such that for any $a\in \mathbb{C}$, we have that $(F^n{)}_{\ast }({\delta }_a)$ converges to a probability measure ${\mu }_F$ for which $F_{\ast }({\mu }_F)={\mu }_{F}d$ where $d$ is the degree of $F$. This result is used to show that the minimal Hutchinson invariant set in degree $n$ of a large class of operators and for sufficiently large $n$ exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, this set is a Cantor set.