On cubic Blaschke products

TL;DR

This paper addresses the dynamics of cubic Blaschke products on the unit disk by establishing a normal form and a structure theorem: every cubic Blaschke product has a unique hyperbolic inflection point in $\mathbb{D}$, which lies at the hyperbolic midpoint of its two critical points. Using hyperbolic divided differences and the Schur–Cohn algorithm, the authors derive an explicit two-parameter polynomial $P(r,s)$ whose sign determines whether a cubic Blaschke product is elliptic, parabolic, or hyperbolic. The degree-two case is reconciled with existing results via a nephroid boundary, while the degree-three case yields a complete parameter-space classification with a parabolic locus given by $P(r,s)=0$ and nephroid-like images for the elliptic/hyperbolic regions. The work advances understanding of Blaschke-product dynamics in the disk and suggests natural generalizations to higher degrees and more complex parameter spaces.

Abstract

We show that every cubic Blaschke product has a unique hyperbolic inflection point in the unit disk and, moreover, this point lies at the hyperbolic midpoint of the two critical points. Using this structure result for cubic Blaschke products, we give an explicit expression in terms of the parameters which determines when cubic Blaschke products are elliptic, parabolic, or hyperbolic.

Figures (6)

• Figure 1: The parameter space for unicritical degree $3$ Blaschke products. The nephroid curve corresponds to the parabolic parameters.
• Figure 2: The parameter space for unicritical degree $3$ polynomials. Observe the nephroid appears here too.
• Figure 3: The slice in parameter space given by $s=-0.8$.
• Figure 4: The slice in parameter space given by $s=0.6$.
• Figure 5: The slice in parameter space given by $s=-0.6i$.

Theorems & Definitions (16)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Gauss-Lucas Theorem
• Theorem 2.2: Denjoy-Wolff Theorem
• Definition 2.3
• Theorem 2.4: Theorem 2.4, Riv11
• Definition 2.5: Definition 3.1, Riv11
• Definition 2.6
• Theorem 2.7: Theorem 3.4, Riv11