In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in detail in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T, we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M^T_CH and find explicitly what operators T have the property that M^T_CH=C.

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_∗⁡(μF)=μ_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.

A linear differential operator $T=Q(z)\frac{d}{dz}+P(z)$ with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, $T$ has a unique minimal Hutchinson-invariant set $M_{CH}^{T}$ in the complex plane. Using a geometric interpretation of its boundary  in terms of envelops of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field $\frac{Q(z)}{P(z)}\partial_{z}$), and singular points of different types which we classify below.\parThe latter decomposition of the boundary of $M_{CH}^{T}$ is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the  field $\frac{Q(z)}{P(z)}\partial_{z}$. We provide an upper bound for the number of local arcs in terms of degrees of $P$ and $Q$. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.

 In this paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic    correspondences. We show that if the critical exponent satisfies $1\leq \deltacrit(x) <+\infty,$ the correspondence $F$ is relatively hyperbolic on the limit set $\Lambda_+(x)$, and $\Lambda_+(x)$ is minimal, then $\Lambda_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff dimension of $\Lambda_+(x)$ is strictly less than $2$. As a special case, this shows that for a parameter $a$ in the interior of a hyperbolic component of the modular Mandelbrot set, the limit set of the Bullett--Penrose correspondence $F_a$ has a non-atomic conformal measure and its Hausdorff dimension is strictly less than $2$. The same results hold for the LLMM correspondences, under some extra assumptions on their defining function $f$. 

This paper, being the sequel of \cite{AlBrSh1}, studies a class of linear ordinary differential operators with polynomial coefficients called exactly solvable; such an operator sends every polynomial of sufficiently large degree to a polynomial of the same degree.

We focus on invariant subsets of the complex plane for such operators when their actionis restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multi-valued counterpart.

As a very special case of invariant sets we recover the Julia sets of rational functions.