An inverse problem in Pólya–Schur theory II. Exactly solvable operators and complex dynamics

Abstract

This paper continues the study of an inverse problem in the Pólya–Schur theory and considers 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 action is restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multivalued counterpart. As a special case of invariant sets, we recover the Julia sets of rational functions.