Matings and the other side of the dictionary
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. S5, pp. 1139-1147.

In the theory of rational maps an important role is played by matings. These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of double limits, that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and Peano curves. Clearly these two constructions form one entry of the Sullivan dictionary. This article attempts to spell out the similarities and differences.

Les accouplements forment une classe essentielle d’applications rationelles, sans doute celle qui est la mieux comprise. Mais elle fait intervenir des objets bizarres : recollements de dendrites, courbes de Peano, etc. La construction analogue pour les groupes Kleiniens est celle des limites doubles. Cette construction est essentielle pour l’hyperbolisation des variétés de dimension trois fibres sur le cercle. Ces deux constructions se correspondent par le dictionnaire de Sullivan. Cet article essaie de montrer les similitudes et les différences.

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Hubbard, John. Matings and the other side of the dictionary. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. S5, pp. 1139-1147. doi : 10.5802/afst.1364. http://www.numdam.org/articles/10.5802/afst.1364/

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