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

TL;DR

This paper introduces the concept of $ ext{ extalpha}$-symmetry as a combinatorial criterion to determine when the mating of two postcritically finite cubic polynomials with a fixed critical point is obstructed, supported by analysis of rotation sets and limb structures.

Contribution

It establishes $ ext{ extalpha}$-symmetry as a sufficient (and conjecturally necessary) condition for mating obstructions in a specific class of cubic polynomials, with proofs and examples.

Findings

$ ext{ extalpha}$-symmetry is sufficient for mating obstruction.

Rotation sets help identify ray classes with closed loops.

Necessity of $ ext{ extalpha}$-symmetry proven for certain maps.

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.