Lavaurs algorithm for cubic symmetric polynomials

TL;DR

This paper develops an algorithm inspired by Lavaurs' method to construct a combinatorial model of the cubic symmetric connectedness locus, enhancing understanding of the parameter space of cubic symmetric polynomials.

Contribution

It introduces a novel algorithm for constructing the lamination $C_sCL$, providing a combinatorial model for the cubic symmetric connectedness locus.

Findings

The algorithm successfully constructs the lamination $C_sCL$.

The model offers a new perspective on the structure of cubic symmetric polynomials.

It parallels the Lavaurs algorithm for quadratic polynomials.

Abstract

To investigate the degree $d$ connectedness locus, Thurston 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 $f_c(z) = z^2 +c$. In the same spirit, we consider the space of all \emph{cubic symmetric polynomials} $f_\lambda(z)=z^3+\lambda^2 z$ in three articles. In the first one we construct the 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, $\mathbb{S}/C_sCL$ is a monotone model of the \emph{cubic symmetric connectedness locus}, i.e., the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for constructing $C_sCL$ analogous to the Lavaurs algorithm for constructing a combinatorial model $\mathcal{M}^{comb}_2$ of the Mandelbrot set $\mathcal{M}_2$.