Lavaurs algorithm for cubic symmetric 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 $C_{s}CL$ together with the induced factor space $\mathbb{S}/C_{s}CL$ of the unit circle $\mathbb{S}$. As will be verified in the third paper, $\mathbb{S}/C_{s}CL$ 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 $C_{s}CL$ analogous to the Lavaurs algorithm for constructing a combinatorial model ${\mathcal{M}}_2^{comb}$ of the Mandelbrot set ${\mathcal{M}}_2$.

27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2201.11434