Geography of the cubic connectedness locus : intertwining surgery
A classification of bicritical rational maps with a pair of period two superattracting cycles
We give a Thurston classification of those bicritical rational maps which have two period two superattracting cycles. We also show that all such maps are constructed by the mating of two unicritical degree polynomials.
Schwarzian derivatives of topologically finite meromorphic functions.
A parabolic Pommerenke-Levin-Yoccoz inequality
In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only...
Twisted matings and equipotential gluings
One crucial tool for studying postcritically finite rational maps is Thurston’s topological characterization of rational maps. This theorem is proved by iterating a holomorphic endomorphism on a certain Teichmüller space. The graph of this endomorphism covers a correspondence on the level of moduli space. In favorable cases, this correspondence is the graph of a map, which can be used to study matings. We illustrate this by way of example: we study the mating of the basilica with itself.
Questions about Polynomial Matings
We survey known results about polynomial mating, and pose some open problems.
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