Symmetric cubic polynomials

TL;DR

This paper introduces a model for the boundary of the connectedness locus of cubic symmetric polynomials and explores the conditions under which a homeomorphism exists between this boundary and the model.

Contribution

It constructs a combinatorial model for the boundary of the cubic symmetric connectedness locus and establishes conditions for a homeomorphism with the actual boundary.

Findings

Existence of a model $ ext{M}_3^{comb}$ for the boundary of the connectedness locus.

A monotone continuous function $ ext{pi}$ maps the boundary to the model.

Homeomorphism holds if the locus is locally connected.

Abstract

We describe a model $\mathcal{M}_3^{comb}$ for the boundary of the connectedness locus $\mathcal{M}^{sy}_3$ of the parameter space of cubic symmetric polynomials $p_c(z)=z^3-3c^2z$. We show that there exists a monotone continuous function $\pi:\partial \mathcal{M}_c^{sy}\to \mathcal{M}_3^{comb}$ which is a homeomorphism if $\mathcal{M}^{sy}_3$ is locally connected.