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Questions about Polynomial Matings
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. S5, pp. 1149-1176.

Nous survolons des résultats connus sur l’accouplement de polynômes et posons quelques problèmes ouverts.

We survey known results about polynomial mating, and pose some open problems.

@article{AFST_2012_6_21_S5_1149_0,
     author = {Xavier Buff and Adam L. Epstein and Sarah Koch and Daniel Meyer and Kevin Pilgrim and Mary Rees and Tan Lei},
     title = {Questions about Polynomial Matings},
     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 = {1149-1176},
     doi = {10.5802/afst.1365},
     zbl = {06167104},
     mrnumber = {3088270},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2012_6_21_S5_1149_0/}
}
Xavier Buff; Adam L. Epstein; Sarah Koch; Daniel Meyer; Kevin Pilgrim; Mary Rees; Tan Lei. Questions about Polynomial Matings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. S5, pp. 1149-1176. doi : 10.5802/afst.1365. https://afst.centre-mersenne.org/item/AFST_2012_6_21_S5_1149_0/

[1] Aspenberg (M.) & Yampolsky (M.).— Mating non-renormalizable quadratic polynomials, Commun. Math. Phys. 287, p. 1-40 (2009). | MR 2480740 | Zbl 1187.37065

[2] Brock (J.), Canary (R.), and Minsky (Y.).— The classification of Kleinian surface groups II: the ending lamination conjecture, To appear, Annals of Mathematics. | MR 2925381 | Zbl pre06074012

[3] Buff (X.), Epstein (A.L.) & Koch (S.).— Twisted matings and equipotential gluing, in this volume.

[4] Blé (G.) & Valdez (R.).— Mating a Siegel disk with the Julia set of a real quadratic polynomial, Conform. Geom. Dyn. 10, p. 257-284 (electronic) (2006). | MR 2261051 | Zbl 1185.37104

[5] Bers (L.).— Simultaneous uniformization, Bull. Amer. Math. Soc. 66, p. 94-97 (1960). | MR 111834 | Zbl 0090.05101

[6] Bullett (S.).— Matings in holomorphic dynamics, in Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser. 368, p. 88-119. Cambridge Univ. Press, Cambridge (2010). | MR 2665006 | Zbl 1206.30002

[7] Cannon (J.) and Thurston (W.).— Group invariant Peano curves, Geometry and Topology 11, p. 1315-1355 (2007). | MR 2326947 | Zbl 1136.57009

[8] Chéritat (A.).— Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle, in this volume.

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

[10] Dudko (D.).— Matings with laminations, arXiv:1112.4780

[11] Epstein (A.).— Quadratic mating discontinuity, manuscript (2012).

[12] Exall (F.).— Rational maps represented by both rabbit and aeroplane matings, PhD thesis, University of Liverpool (2011).

[13] Hruska Boyd (S.).— The Medusa algorithm for polynomial matings, arXiv:1102.5047.

[14] Hubbard (J.).— Matings and the other side of the dictionary, in this volume.

[15] Hubbard (J.).— Preface, in The Mandelbrot set, Theme and Variations, London Math. Soc. Lecture Note Series 274, p. xiii-xx. Cambridge University Press (2000). | MR 1765081 | Zbl 1107.37304

[16] Haïssinsky (P.) & Tan (L.).— Convergence of pinching deformations and matings of geometrically finite polynomials, Fund. Math. 181, p. 143-188. | MR 2070668 | Zbl 1048.37045

[17] Kameyama (A.).— On Julia sets of postcritically finite branched coverings. II. S1-parametrization of Julia sets. J. Math. Soc. Japan 55, p. 455-468 (2003). | MR 1961296 | Zbl 1162.37319

[18] Kiwi (J.) & Rees (M.).— Counting hyperbolic components, submitted to the London Mathematical Society.

[19] Luo (J.).— Combinatorics and holomorphic dynamics: Captures, matings, Newton’s method, Ph.D. Thesis, Cornell University (1995). | MR 2691795

[20] Minsky (Y.).— On Thurston’s ending lamination conjecture, in Low-dimensional topology (Knoxville, TN, 1994), Conf. Proc. Lecture Notes Geom. Topology, III, p. 109-122. Int. Press, Cambridge, MA (1994). | MR 1316176 | Zbl 0846.57010

[21] Mashanova (I.) & Timorin (V.).— Captures, matings, and regulings, arxiv:1111.5696.

[24] Meyer (D.).— Expanding Thurston maps as quotients, .

[25] Meyer (D.).— Invariant Peano curves of expanding Thurston maps, to appear, Acta. Math., .

[26] Meyer (D.).— Unmating of rational maps, sufficient criteria and examples, arXiv:1110.6784, (2011), to appear in the Proc. to Milnor’s 80th birthday.

[27] Meyer (D.) & Petersen (C.).— On the notions of matings, in this volume.

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

[29] Milnor (J.) and Tan (L.).— Remarks on quadratic rational maps (with an appendix by Tan Lei), Experimental Math 2, p. 37-83 (1993). | MR 1246482 | Zbl 0922.58062

[30] Mj (M.).— Cannon-Thurston maps for surface groups II: split geometry and the Minsky model, , preprint; accessed June 11 (2012).

[31] Mj (M.).— Cannon-Thurston maps for surface groups, , preprint.

[32] Petersen (C.).— No elliptic limits for quadratic rational maps, Ergodic Theory Dynam. Systems 19, p. 127-141 (1999). | MR 1676926 | Zbl 0921.30019

[33] Rees (M.).— Realization of matings of polynomials of rational maps of degree two, Manuscript (1986).

[34] Rees (M.).— Components of degree two hyperbolic rational maps, Invent. Math., 100, p. 357-382 (1990). | MR 1047139 | Zbl 0712.30022

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

[36] Rees (M.).— Multiple equivalent matings with the aeroplane polynomial, Erg. Th. and Dyn. Sys., 30, p. 1239-1257 (2010). | MR 2669420 | Zbl pre05772361

[37] Sharland (T.).— Rational Maps with Clustering and the Mating of Polynomials, PhD thesis, Warwick (2010).

[38] Sharland (T.).— Constructing rational maps with cluster points using the mating operation, Preprint (2011).

[39] Shishikura (M.).— On a theorem of M. Rees for matings of polynomials, London Math. Soc. Lecture Note Ser., 274. CMP 2000:14. | MR 1765095 | Zbl 1062.37039

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

[41] Tan (L.).— Branched coverings and cubic Newton maps, Fund. Math. 154, p. 207-260 (1997). | MR 1475866 | Zbl 0903.58029

[42] Tan (L.).— Matings of quadratic polynomials, Erg. Th. and Dyn. Sys. 12, p. 589-620 (1992). | MR 1182664 | Zbl 0756.58024

[43] Tan (L.).— On pinching deformations of rational maps, Ann. Sci. École Norm. Sup. 35, p. 353-370 (2002). | Numdam | MR 1914001 | Zbl 1041.37022

[44] Wittner (B.).— On the bifurcation loci of rational maps of degree two, Ph.D. thesis, Cornell University (1988). | MR 2636558

[45] Yampolsky (M.) & Zakeri (S.).— Mating Siegel quadratic polynomials, Journ. of the A.M.S., vol 14-1, p. 25-78 (2000). | MR 1800348 | Zbl 1050.37022

[46] Zhang (G.).— All David type Siegel disks of polynomial maps are Jordan domains, manuscript (2009).