Thurston obstructions for cubic branched coverings with two fixed critical points

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any obstruction for $F$ contains a Levy cycle. This note is part of a study by the author investigating the mating of pairs of cubic polynomials which each have a fixed critical point.Comment: 8 page note, part of study of matings of cubic polynomials. Updated to modify statement of main theorem and clarify a remar

Newton maps as matings of cubic polynomials

International audienceIn this paper we prove existence and uniqueness of matings of a large class of renormalizable cubic polynomials with one fixed critical point and the other cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our result is the first part towards a conjecture by Tan Lei, stating that all (cubic) Newton maps can be described as matings or captures

Constructing rational maps with cluster points using the mating operation

In this article, we show that all admissible rational maps with fixed or period two cluster cycles can be constructed by the mating of polynomials. We also investigate the polynomials which make up the matings that construct these rational maps. In the one cluster case, one of the polynomials must be an $n$-rabbit and in the two cluster case, one of the maps must be either $f$, a "double rabbit", or $g$, a secondary map which lies in the wake of the double rabbit $f$. There is also a very simple combinatorial way of classifiying the maps which must partner the aforementioned polynomials to create rational maps with cluster cycles. Finally, we also investigate the multiplicities of the shared matings arising from the matings in the paper.Comment: 23 page

Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle

After giving an introduction to the procedure dubbed slow polynomial mating and stating a conjecture relating this to other notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.Comment: 35 page