Epicycloids and Blaschke products

TL;DR

This paper demonstrates that the parameter space of degree d unicritical Blaschke products features an epicycloid with d-1 cusps, analogous to the Multibrot set, and characterizes parameters with attracting fixed points.

Contribution

It extends the known epicycloid boundary characterization from Multibrot sets to unicritical Blaschke products, providing a new geometric insight.

Findings

Parabolic parameters form an epicycloid with d-1 cusps.

Parameters inside the epicycloid correspond to elliptic functions with attracting fixed points.

Special case analysis for degree 2 Blaschke products.

Abstract

It is well known that the bounding curve of the central hyperbolic component of the Multibrot set in the parameter space of unicritical degree $d$ polynomials is an epicycloid with $d-1$ cusps. The interior of the epicycloid gives the polynomials of the form $z^d+c$ which have an attracting fixed point. We prove an analogous result for unicritical Blaschke products: in the parameter space of degree $d$ unicritical Blaschke products, the parabolic functions are parameterized by an epicycloid with $d-1$ cusps and inside this epicycloid are the parameters which give rise to elliptic functions having an attracting fixed point in the unit disk. We further study in more detail the case when $d=2$ in which every Blaschke product is unicritical in the unit disk.