Symmetric cubic laminations

Abstract

To investigate the degree d 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 σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z 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 C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \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.