EQUIDISTRIBUTION OF ITERATIONS OF HOLOMORPHIC CORRESPONDENCES AND HUTCHINSON INVARIANT SETS

In this paper, we analyze a certain family of holomorphic correspondences on Ĉ×Ĉ 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 ∈ C, we have that (Fⁿ)∗(δ_a) converges to a probability measure μ_F for which F_∗(μ_F) = μ_Fd where d is the degree of 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 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.