Linear first order differential operators and their Hutchinson-invariant sets

TL;DR

This paper explores the relationship between linear first order differential operators with polynomial coefficients and complex dynamics, introducing Hutchinson-invariant sets and analyzing their properties.

Contribution

It introduces the concept of Hutchinson-invariant sets for linear differential operators and characterizes their properties, including the existence of a unique minimal invariant set.

Findings

Existence of a unique minimal Hutchinson-invariant set for non-constant coefficient operators.

Explicit conditions under which the invariant set equals the entire complex plane.

Development of a new interpretation linking differential operators with complex dynamics.

Abstract

In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details 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 $\mathrm{M}^T_{CH}$ and find explixitly when it equals $\mathbb{C}$.