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Abstract As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so calledquadratic minor laminationof Thurston. In the same spirit, we consider the space of allcubic symmetric polynomials$$f_\unicode{x3bb} (z)=z^3+\unicode{x3bb} ^2 z$$in three articles. In the first one, we construct thecubic symmetric comajor laminationtogether with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of thecubic symmetric connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination. 

Abstract We describe a locally connected model of the cubic connectedness locus. The model is obtained by constructing a decomposition of the space of critical portraits and collapsing elements of the decomposition into points. This model is similar to a quotient of the combinatorial quadratic Mandelbrot set in which all baby Mandelbrot sets, as well as the filled Main Cardioid, are collapsed to points. All fibres of the model, possibly except one, are connected. The authors are not aware of other known models of the cubic connectedness locus. 

Abstract The modulus of a polynomial-like (PL) map is an important invariant that controls distortion of the straightening map and, hence, geometry of the corresponding PL Julia set. Lower bounds on the modulus, calledcomplex a priori bounds, are known in a great variety of contexts. For any rational function we complement this by an upper bound for moduli of PL maps in the satellite case that dependsonly on the relative period and the degree of the PL map.This rules out a priori bounds in the satellite case with unbounded relative periods. We also apply our tools to obtain lower bounds for hyperbolic lengths of geodesics in the infinitely renormalizable case, and to show that moduli of annuli must converge to 0 for a sequence of arbitrary renormalizations, under several conditions all of which are shown to be necessary.