Unicritical Blaschke products and domains of ellipticity

TL;DR

This paper investigates the domain of ellipticity for unicritical Blaschke products, characterizing the parameter space and relating it to the Mandelbrot set, revealing how the elliptic domain extends to include a specific boundary point.

Contribution

It provides a detailed analysis of the elliptic parameter domain for unicritical Blaschke products and connects it to the Mandelbrot set, highlighting a new geometric relationship.

Findings

Characterization of the elliptic domain as a subset of the parameter space.

Identification of the Mandelbrot set analogue for this family.

The elliptic domain is extended by adding a single boundary point.

Abstract

Elliptic M\"obius transformations of the unit disk are those for which there is a fixed point in $\mathbb{D}$. It is not hard to classify which M\"obius transformations are elliptic in terms of the parameters. The set of parameters can be identified with the solid torus $S^1 \times \mathbb{D}$, and the set of elliptic parameters is called the domain of ellipticity. In this paper, we study the domain of ellipticity for non-trivial unicritical Blaschke products. We will also study the set corresponding to the Mandelbrot set for this family, and show how it can be obtained from the domain of ellipticity by adding one point.