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arXiv · 2205.03506

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

Thomas Sharland
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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.

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Thomas Sharland. 2022-05-06. Matings of cubic polynomials with a fixed critical point. Part II: $\alpha$-symmetry of limbs. https://doi.org/10.1080/10236198.2023.2188973

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