Questions about Polynomial Matings

Xavier Buff[1]; Adam L. Epstein[2]; Sarah Koch[3]; Daniel Meyer[4]; Kevin Pilgrim[4]; Mary Rees[5]; Tan Lei[6]

  • [1] Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France
  • [2] Mathematics institute, University of Warwick, Coventry CV4 7AL, United Kingdom
  • [3] Department of Mathematics, Science Center, 1 Oxford Street, Harvard University, Cambridge MA 02138, United States
  • [4] Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany
  • [5] Department of Mathematical Sciences, University of Liverpool, Mathematics Building  Peach St., Liverpool L69 7ZL U.K.
  • [6] LUNAM Université, Université d’Angers, LAREMA – Laboratoire Angevin de Recherche en Mathématiques, 2 bd Lavoisier, 49045 Angers, cedex, France

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: S5, page 1149-1176
  • ISSN: 0240-2963

Abstract

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

How to cite

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Buff, Xavier, et al. "Questions about Polynomial Matings." Annales de la faculté des sciences de Toulouse Mathématiques 21.S5 (2012): 1149-1176. <http://eudml.org/doc/251009>.

@article{Buff2012,
abstract = {We survey known results about polynomial mating, and pose some open problems.},
affiliation = {Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France; Mathematics institute, University of Warwick, Coventry CV4 7AL, United Kingdom; Department of Mathematics, Science Center, 1 Oxford Street, Harvard University, Cambridge MA 02138, United States; Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany; Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany; Department of Mathematical Sciences, University of Liverpool, Mathematics Building  Peach St., Liverpool L69 7ZL U.K.; LUNAM Université, Université d’Angers, LAREMA – Laboratoire Angevin de Recherche en Mathématiques, 2 bd Lavoisier, 49045 Angers, cedex, France},
author = {Buff, Xavier, Epstein, Adam L., Koch, Sarah, Meyer, Daniel, Pilgrim, Kevin, Rees, Mary, Lei, Tan},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {12},
number = {S5},
pages = {1149-1176},
publisher = {Université Paul Sabatier, Toulouse},
title = {Questions about Polynomial Matings},
url = {http://eudml.org/doc/251009},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Buff, Xavier
AU - Epstein, Adam L.
AU - Koch, Sarah
AU - Meyer, Daniel
AU - Pilgrim, Kevin
AU - Rees, Mary
AU - Lei, Tan
TI - Questions about Polynomial Matings
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/12//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - S5
SP - 1149
EP - 1176
AB - We survey known results about polynomial mating, and pose some open problems.
LA - eng
UR - http://eudml.org/doc/251009
ER -

References

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  1. Aspenberg (M.) & Yampolsky (M.).— Mating non-renormalizable quadratic polynomials, Commun. Math. Phys. 287, p. 1-40 (2009). Zbl1187.37065MR2480740
  2. Brock (J.), Canary (R.), and Minsky (Y.).— The classification of Kleinian surface groups II: the ending lamination conjecture, To appear, Annals of Mathematics. Zbl1253.57009MR2925381
  3. Buff (X.), Epstein (A.L.) & Koch (S.).— Twisted matings and equipotential gluing, in this volume. Zbl06167099
  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). Zbl1185.37104MR2261051
  5. Bers (L.).— Simultaneous uniformization, Bull. Amer. Math. Soc. 66, p. 94-97 (1960). Zbl0090.05101MR111834
  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). Zbl1206.30002MR2665006
  7. Cannon (J.) and Thurston (W.).— Group invariant Peano curves, Geometry and Topology 11, p. 1315-1355 (2007). Zbl1136.57009MR2326947
  8. Chéritat (A.).— Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle, in this volume. Zbl1320.37023
  9. Douady (A.) & Hubbard (J.H.).— A Proof of Thurston’s characterization of rational functions, Acta. Math. 171, p. 263-297 (1993). Zbl0806.30027MR1251582
  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. Zbl1291.37054
  14. Hubbard (J.).— Matings and the other side of the dictionary, in this volume. Zbl1283.37052
  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). Zbl1107.37304MR1765081
  16. Haïssinsky (P.) & Tan (L.).— Convergence of pinching deformations and matings of geometrically finite polynomials, Fund. Math. 181, p. 143-188. Zbl1048.37045MR2070668
  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). Zbl1162.37319MR1961296
  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). MR2691795
  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). Zbl0846.57010MR1316176
  21. Mashanova (I.) & Timorin (V.).— Captures, matings, and regulings, arxiv:1111.5696. 
  22. Meyer (D.).— Expanding Thurston maps as quotients, http://arxiv.org/abs/0910.2003. 
  23. Meyer (D.).— Invariant Peano curves of expanding Thurston maps, to appear, Acta. Math., http://arxiv.org/abs/0907.1536. Zbl1333.37043
  24. Meyer (D.).— Unmating of rational maps, sufficient criteria and examples, arXiv:1110.6784, (2011), to appear in the Proc. to Milnor’s 80th birthday. Zbl06490013
  25. Meyer (D.) & Petersen (C.).— On the notions of matings, in this volume. 
  26. Milnor (J.).— Pasting together Julia sets; a worked out example of mating, Experimental Math 13 p. 55-92 (2004). Zbl1115.37051MR2065568
  27. Milnor (J.) and Tan (L.).— Remarks on quadratic rational maps (with an appendix by Tan Lei), Experimental Math 2, p. 37-83 (1993). Zbl0922.58062MR1246482
  28. 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). 
  29. Mj (M.).— Cannon-Thurston maps for surface groups, http://arxiv.org/pdf/math.GT/0607509.pdf, preprint. Zbl1301.57013
  30. Petersen (C.).— No elliptic limits for quadratic rational maps, Ergodic Theory Dynam. Systems 19, p. 127-141 (1999). Zbl0921.30019MR1676926
  31. Rees (M.).— Realization of matings of polynomials of rational maps of degree two, Manuscript (1986). 
  32. Rees (M.).— Components of degree two hyperbolic rational maps, Invent. Math., 100, p. 357-382 (1990). Zbl0712.30022MR1047139
  33. Rees (M.).— A partial description of parameter space of rational maps of degree two: part I, Acta Math., 168 p. 11-87 (1992). Zbl0774.58035MR1149864
  34. Rees (M.).— Multiple equivalent matings with the aeroplane polynomial, Erg. Th. and Dyn. Sys., 30, p. 1239-1257 (2010). Zbl1291.37069MR2669420
  35. Sharland (T.).— Rational Maps with Clustering and the Mating of Polynomials, PhD thesis, Warwick (2010). 
  36. Sharland (T.).— Constructing rational maps with cluster points using the mating operation, Preprint (2011). Zbl1333.37033
  37. Shishikura (M.).— On a theorem of M. Rees for matings of polynomials, London Math. Soc. Lecture Note Ser., 274. CMP 2000:14. Zbl1062.37039MR1765095
  38. Shishikura (M.) & Tan (L.).— A family of cubic rational maps and matings of cubic polynomials, Experiment. Math. 9, p. 29-53 (2000). Zbl0969.37020MR1758798
  39. Tan (L.).— Branched coverings and cubic Newton maps, Fund. Math. 154, p. 207-260 (1997). Zbl0903.58029MR1475866
  40. Tan (L.).— Matings of quadratic polynomials, Erg. Th. and Dyn. Sys. 12, p. 589-620 (1992). Zbl0756.58024MR1182664
  41. Tan (L.).— On pinching deformations of rational maps, Ann. Sci. École Norm. Sup. 35, p. 353-370 (2002). Zbl1041.37022MR1914001
  42. Wittner (B.).— On the bifurcation loci of rational maps of degree two, Ph.D. thesis, Cornell University (1988). MR2636558
  43. Yampolsky (M.) & Zakeri (S.).— Mating Siegel quadratic polynomials, Journ. of the A.M.S., vol 14-1, p. 25-78 (2000). Zbl1050.37022MR1800348
  44. Zhang (G.).— All David type Siegel disks of polynomial maps are Jordan domains, manuscript (2009). 

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