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

Abstract

Following Milnor, define to be the space of monic, centred cubic polynomials with a marked fixed critical point. In this article, a sequel to [Thomas Sharland, Matings of cubic polynomials with a fixed critical point, part I: Thurston obstructions, Conform. Geom. Dyn. 23 (12) (2019), pp. 205–220], we provide a combinatorial sufficient (and conjecturally, necessary) condition (called α-symmetry) for the mating of two postcritically finite polynomials in to be obstructed. To do this, we study the rotation sets associated to the parameter limbs in the connectedness locus of , 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 α-symmetry for a particular subset of postcritically finite maps in . Many examples are given to illustrate the results of the paper.