Symmetric cubic polynomials

Abstract

We describe a model ${\mathcal{M}}_3^{comb}$ for the boundary of the connectedness locus ${\mathcal{M}}_3^{sy}$ of the parameter space of cubic symmetric polynomials $p_c(z)=z^3-3c^{2}z$. 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 ${\mathcal{M}}_3^{sy}$ is locally connected.

34 pages, 3 figures