Classification of polynomials and an orthonormal basis construction on the associated basin of attraction

Abstract

We study a construction that associates certain Fatou subsets with reproducing kernel Hilbert spaces and a method for constructing an orthonormal basis for said Hilbert space, which depends on the map of the given Fatou set. We provide a classification of those polynomials the method applies to, and we extend a previous result concerning the dynamics of these orthonormal bases. Essential to the proof of this classification are the Newton--Girard identities and Vieta's formulas.