Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets

In this paper, we analyze a certain family of holomorphic correspondences on C ^ × C ^ Ĉ× Ĉ and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map F F is such that for any a ∈ C a∈ C , we have that ( F n ) ∗ ( δ a ) (F^n)_*(δ _a) converges to a probability measure μ F μ _F for which F ∗ ( μ F ) = μ F d F_*(μ _F)=μ _F d where d d is the degree of F F . This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [An inverse problem in Pólya–Schur theory. I. Non-degenerate and degenerate operators, preprint, 2024], of a large class of operators and for sufficiently large n n exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set.