Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets

In this paper, we analyze a certain family of holomorphic correspondences on C ^ × C ^ {C} {C} 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) (Fⁿ) _* (ₐ) converges to a probability measure μ F F for which F ∗ (μ F) = μ F d F_* (F) = 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.