Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. S5, p. 935-980

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.

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.

     author = {Ch\'eritat, Arnaud},
     title = {Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 21},
     number = {S5},
     year = {2012},
     pages = {935-980},
     doi = {10.5802/afst.1358},
     mrnumber = {3088263},
     zbl = {06167097},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2012_6_21_S5_935_0}
Chéritat, Arnaud. Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. S5, pp. 935-980. doi : 10.5802/afst.1358. http://www.numdam.org/item/AFST_2012_6_21_S5_935_0/

[1] Buff (X.), Fehrenbach (J.), Lochak (P.), Schneps (L.) and Vogel (P.).— Moduli spaces of curves, mapping class groups and fields theory, volume 9 of SMF/AMS Texts and Monographs. American Mathematical Society, Providence, RI (2003). Translated from the French by Schneps. | MR 2006093 | Zbl 1024.32010

[2] Buff (X.), Epstein (A. L.) and Koch (S.).— Twisted matings and equipotential gluings, Ann. Fac. Sci. Toulouse Math (6), 21(5), (2012).

[3] Buff (X.), Epstein (A. L.), Meyer (D.), Pilgrim (K. M.), Rees (M.) and Tan (L.).— Questions about polynomial matings, Ann. Fac. Sci. Toulouse Math (6), 21(5), (2012).

[4] Douady (A.) and Hubbard (J. H.).— A proof of Thurston’s topological characterization of rational functions. Acta Math., 171(2) p. 263-297 (1993). | MR 1251582 | Zbl 0806.30027

[5] Hubbard (J. H.).— Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1. Matrix Editions, Ithaca, NY (2006). Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle. | MR 2245223 | Zbl 1102.30001

[6] Koch (S.).— A new link between Teichmüller theory and complex dynamics (PhD thesis). PhD thesis, Cornell Univ. (2008).

[7] Levy (S.).— Critically Finite Rational Maps (PhD thesis). PhD thesis, Princeton Univ. (1985).

[8] Meyer (D.), Petersen (C. L.).— On the Notion of Matings, Ann. Fac. Sci. Toulouse Math (6), 21(5), (2012).

[9] Milnor (J.).— Geometry and dynamics of quadratic rational maps. Experiment. Math., 2(1) p. 37-83 (1993). With an appendix by the author and Lei Tan. | MR 1246482 | Zbl 0922.58062

[10] Milnor (J.).— Pasting together Julia sets: a worked out example of mating. Experiment. Math., 13(1) p. 55-92 (2004). | MR 2065568 | Zbl 1115.37051

[11] Pilgrim (K. M.).— Canonical Thurston obstructions. Adv. Math., 158(2) p. 154-168 (2001). | MR 1822682 | Zbl 1193.57002

[12] Pilgrim (K.).— An algebraic formulation of Thurston’s characterization of rational functions, Ann. Fac. Sci. Toulouse Math (6), 21(5), (2012). | Zbl 1272.37025

[13] Rees (M.).— A partial description of parameter space of rational maps of degree two. I. Acta Math., 168(1-2) p. 11-87 (1992). | MR 1149864 | Zbl 0774.58035

[14] Selinger (N.).— Thurston’s pullback map on the augmented Teichmüller space and applications. arXiv:1010.1690v2. | Zbl 1298.37033

[15] Shishikura (M.).— On a theorem of M. Rees for matings of polynomials. In The Mandelbrot set, theme and variations, volume 274 of London Math. Soc. Lecture Note Ser., p. 289-305. Cambridge Univ. Press, Cambridge (2000). | MR 1765095 | Zbl 1062.37039

[16] Shishikura (M.) and Tan (L.).— A family of cubic rational maps and matings of cubic polynomials. Experiment. Math., 9(1) p. 29-53 (2000). | MR 1758798 | Zbl 0969.37020