Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets

Abstract:

In this paper, we analyze a certain family of holomorphic correspondences on $\hat {\mathbb {C}}\times \hat {\mathbb {C}}$ 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 $(F^n)_*(\delta _a)$ converges to a probability measure $\mu _F$ for which $F_*(\mu _F)=\mu _F d$ 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.