On boundary points of minimal continuously Hutchinson invariant sets

TL;DR

This paper studies the boundary structure of minimal Hutchinson-invariant sets generated by a polynomial-coefficient differential operator, classifying boundary points and providing geometric insights and bounds related to the operator's coefficients.

Contribution

It introduces a geometric interpretation of the boundary of these invariant sets, classifies boundary points, and establishes bounds based on polynomial degrees, advancing understanding of Hutchinson invariants.

Findings

Boundary boundary points are classified into local and global arcs.

An upper bound on the number of local arcs is provided based on polynomial degrees.

Global geometric properties of minimal Hutchinson-invariant sets are derived.

Abstract

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. The 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.