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

Chéritat, Arnaud

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro Spécial à l’occasion du “Workshop on polynomial matings” 8-11 juin 2011, Toulouse, Tome 21 (2012) no. S5, pp. 935-980.

Après avoir donné une introduction à la procédure baptisée accouplement lent de polynômes et avoir rapidement rappelé des résultats connus sur la notion plus classique d’accouplement de polynômes, nous montrons des images conformément correctes de l’accouplement lent de deux polynômes de degré $3$ post critiquement finis introduits par Shishikura et Tan Lei en tant qu’exemple de paire obstruée mais sans cycle de Levy. Ces images semblent montrer que les ensembles de Julia déformés ont une limite, qui semble reliée à l’ensemble de Julia d’une application rationnelle de degré $6$. Nous donnons une interprétation conjecturale de ces faits en termes de sphères pincées et montrons d’autres représentations conformes.

After giving an introduction to the procedure dubbed slow polynomial mating and quickly recalling known results about more classical 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.

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