Untitled

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 σ d -invariant laminations, where σ 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 λ (z) = z 3 + λ 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 S/C s CL of the unit circle S. As will be verified in the third paper of the series, S/C s CL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.