Matings of cubic polynomials with a fixed critical point. Part II: $\alpha$-symmetry of limbs

Abstract

In this article we provide a combinatorial sufficient (and conjecturally, necessary) condition (called $\alpha$-symmetry) for the mating of two postcritically finite polynomials in $\mathcal{S}_1$ to be obstructed. To do this, we study the rotation sets associated to the parameter limbs in the connectedness locus of $\mathcal{S}_1$, which allows us to determine when there exist ray classes in the formal mating which contain a closed loop. We give a proof of the necessity of $\alpha$-symmetry for a particular subset of postcritically finite maps in $\mathcal{S}_1$. Many examples are given to illustrate the results of the paper.