Julia sets of Orthogonal polynomials

TL;DR

This paper explores the relationship between the dynamical properties of orthogonal polynomials and the geometric features of their support in the complex plane, focusing on Julia sets and Green's functions.

Contribution

It establishes connections between Julia sets, filled Julia sets, and Green's functions of orthogonal polynomials and the support's geometric boundary and convex hull.

Findings

Julia set of $P_n$ relates to the support's outer boundary

Filled Julia set corresponds to the polynomial convex hull of the support

Green's function of $P_n$ matches that of the support's complement

Abstract

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials $\{P_n\}$ to properties of the support. More precisely we relate the Julia set of $P_n$ to the outer boundary of the support, the filled Julia set to the polynomial convex hull $K$ of the support, and the Green's function associated with $P_n$ to the Green's function for the complement of $K$.