Symmetric cubic laminations

Abstract:

To investigate the degree $d$ connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied $\sigma _d$-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 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 cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.