An inverse problem in P\'olya--Schur theory. II. Exactly solvable operators and complex dynamics

TL;DR

This paper investigates a class of linear differential operators that preserve polynomial degrees and explores their invariant sets in the complex plane, revealing connections to complex dynamics and Julia sets.

Contribution

It introduces and analyzes exactly solvable operators in Pólya-Schur theory, linking their invariant sets to classical complex dynamics and Julia sets.

Findings

Identification of invariant sets related to Julia sets

Connection between differential operators and complex dynamics

Extension of previous work on Pólya-Schur operators

Abstract

This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called \emph{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 multi-valued counterpart. As a very special case of invariant sets we recover the Julia sets of rational functions.