We survey known results about polynomial mating, and pose some open problems.
Nous survolons des résultats connus sur l’accouplement de polynômes et posons quelques problèmes ouverts.
Xavier Buff 1; Adam L. Epstein 2; Sarah Koch 3; Daniel Meyer 4; Kevin Pilgrim 4; Mary Rees 5; Tan Lei 6
@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},
pages = {1149--1176},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {S5},
year = {2012},
doi = {10.5802/afst.1365},
mrnumber = {3088270},
zbl = {06167104},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1365/}
}
TY - JOUR AU - Xavier Buff AU - Adam L. Epstein AU - Sarah Koch AU - Daniel Meyer AU - Kevin Pilgrim AU - Mary Rees AU - Tan Lei TI - Questions about Polynomial Matings JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 1149 EP - 1176 VL - 21 IS - S5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1365/ DO - 10.5802/afst.1365 LA - en ID - AFST_2012_6_21_S5_1149_0 ER -
%0 Journal Article %A Xavier Buff %A Adam L. Epstein %A Sarah Koch %A Daniel Meyer %A Kevin Pilgrim %A Mary Rees %A Tan Lei %T Questions about Polynomial Matings %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 1149-1176 %V 21 %N S5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1365/ %R 10.5802/afst.1365 %G en %F 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, Serie 6, Volume 21 (2012) no. S5, pp. 1149-1176. doi : 10.5802/afst.1365. https://afst.centre-mersenne.org/articles/10.5802/afst.1365/
[1] Aspenberg (M.) & Yampolsky (M.).— Mating non-renormalizable quadratic polynomials, Commun. Math. Phys. 287, p. 1-40 (2009). | MR | Zbl
[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 | Zbl
[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 | Zbl
[5] Bers (L.).— Simultaneous uniformization, Bull. Amer. Math. Soc. 66, p. 94-97 (1960). | MR | Zbl
[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 | Zbl
[7] Cannon (J.) and Thurston (W.).— Group invariant Peano curves, Geometry and Topology 11, p. 1315-1355 (2007). | MR | Zbl
[8] Chéritat (A.).— Tan Lei and Shishikura’s example of non-mateable degree 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 | Zbl
[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 | Zbl
[16] Haïssinsky (P.) & Tan (L.).— Convergence of pinching deformations and matings of geometrically finite polynomials, Fund. Math. 181, p. 143-188. | MR | Zbl
[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 | Zbl
[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
[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 | Zbl
[21] Mashanova (I.) & Timorin (V.).— Captures, matings, and regulings, arxiv:1111.5696.
[24] Meyer (D.).— Expanding Thurston maps as quotients, . | arXiv
[25] Meyer (D.).— Invariant Peano curves of expanding Thurston maps, to appear, Acta. Math., . | arXiv
[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 | Zbl
[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 | Zbl
[30] Mj (M.).— Cannon-Thurston maps for surface groups II: split geometry and the Minsky model, http://lists.rkmvu.ac.in/intro/academics/matsc_website/mahan/split.pdf, preprint; accessed June 11 (2012).
[31] Mj (M.).— Cannon-Thurston maps for surface groups, , preprint. | arXiv
[32] Petersen (C.).— No elliptic limits for quadratic rational maps, Ergodic Theory Dynam. Systems 19, p. 127-141 (1999). | MR | Zbl
[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 | Zbl
[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 | Zbl
[36] Rees (M.).— Multiple equivalent matings with the aeroplane polynomial, Erg. Th. and Dyn. Sys., 30, p. 1239-1257 (2010). | MR
[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 | Zbl
[40] Shishikura (M.) & Tan (L.).— A family of cubic rational maps and matings of cubic polynomials, Experiment. Math. 9, p. 29-53 (2000). | MR | Zbl
[41] Tan (L.).— Branched coverings and cubic Newton maps, Fund. Math. 154, p. 207-260 (1997). | MR | Zbl
[42] Tan (L.).— Matings of quadratic polynomials, Erg. Th. and Dyn. Sys. 12, p. 589-620 (1992). | MR | Zbl
[43] Tan (L.).— On pinching deformations of rational maps, Ann. Sci. École Norm. Sup. 35, p. 353-370 (2002). | Numdam | MR | Zbl
[44] Wittner (B.).— On the bifurcation loci of rational maps of degree two, Ph.D. thesis, Cornell University (1988). | MR
[45] Yampolsky (M.) & Zakeri (S.).— Mating Siegel quadratic polynomials, Journ. of the A.M.S., vol 14-1, p. 25-78 (2000). | MR | Zbl
[46] Zhang (G.).— All David type Siegel disks of polynomial maps are Jordan domains, manuscript (2009).
Cited by Sources: