Symmetric Cubic Laminations

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 $C_{s}CL$ 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 of the series, $\mathbb{S}/C_{s}CL$ is a monotone model of the \emph{cubic symmetric connected locus}, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

37 pages, 4 figures