Symmetric Cubic Laminations

TL;DR

This paper constructs a lamination model for the space of cubic symmetric polynomials with connected Julia sets, extending Thurston's lamination approach from quadratic to cubic cases.

Contribution

It introduces a new lamination $C_sCL$ and a corresponding factor space that models the cubic symmetric connected locus.

Findings

Constructed lamination $C_sCL$ for cubic symmetric polynomials.

Established the factor space as a monotone model of the connected locus.

Lays groundwork for further topological analysis of cubic polynomial dynamics.

Abstract

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f(z) = z^2 +c$. In the spirit of Thurston's work, we consider the space of all \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in a series of three articles. In the present paper, the first in the series, we construct a lamination $C_sCL$ together with the induced factor space ${\mathbb{S}}/C_sCL$ of the unit circle ${\mathbb{S}}$. As will be verified in the third paper of the series, ${\mathbb{S}}/C_sCL$ is a monotone model of the \emph{cubic symmetric connected locus}, i.e. the space of all cubic symmetric polynomials with connected Julia sets.