Symmetric cubic laminations

Conformal Geometry and Dynamics. 2023. Vol. 27. No. 8. P. 264–293.

Blokh A., Oversteegen L., Selinger N., Timorin V., Vejandla S. C.

To investigate the degree $d$ connectedness locus, Thurston 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 $\mathbb{S}^1/C_sCL$ of the unit circle $\mathbb{S}^1$. As will be verified in the third paper of the series, $\mathbb{S}^1/C_sCL$ is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

Keywords: Julia set, Mandelbrot set, complex dynamics, laminations