Julia sets of Orthogonal polynomials
Jacob Stordahl Christiansen, Christian Henriksen, Henrik Laurberg, Pedersen, Carsten Lunde Petersen

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 to properties of the support. More precisely we relate the Julia set of to the outer boundary of the support, the filled Julia set to the polynomial convex hull of the support, and the Green's function associated with to the Green's function for the complement of .
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Taxonomy
TopicsMathematical Dynamics and Fractals · Functional Equations Stability Results · Chaos control and synchronization
