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

Arnaud Chéritat[1]

  • [1] Centre National de la Recherche Scientifique, Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 Route de Narbonne 31062 Toulouse Cedex 9, France

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

  • Volume: 21, Issue: S5, page 935-980
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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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 21.S5 (2012): 935-980. <http://eudml.org/doc/251012>.

@article{Chéritat2012,
abstract = {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.},
affiliation = {Centre National de la Recherche Scientifique, Institut de Mathématiques de Toulouse, Université Paul Sabatier 118 Route de Narbonne 31062 Toulouse Cedex 9, France},
author = {Chéritat, Arnaud},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {polynomial mating; postcritcally finite},
language = {eng},
month = {12},
number = {S5},
pages = {935-980},
publisher = {Université Paul Sabatier, Toulouse},
title = {Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle},
url = {http://eudml.org/doc/251012},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Chéritat, Arnaud
TI - Tan Lei and Shishikura’s example of non-mateable degree 3 polynomials without a Levy cycle
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/12//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - S5
SP - 935
EP - 980
AB - 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.
LA - eng
KW - polynomial mating; postcritcally finite
UR - http://eudml.org/doc/251012
ER -

References

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  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. Zbl1024.32010MR2006093
  2. Buff (X.), Epstein (A. L.) and Koch (S.).— Twisted matings and equipotential gluings, Ann. Fac. Sci. Toulouse Math (6), 21(5), (2012). Zbl06167099
  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). Zbl0806.30027MR1251582
  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. Zbl1102.30001MR2245223
  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. Zbl0922.58062MR1246482
  10. Milnor (J.).— Pasting together Julia sets: a worked out example of mating. Experiment. Math., 13(1) p. 55-92 (2004). Zbl1115.37051MR2065568
  11. Pilgrim (K. M.).— Canonical Thurston obstructions. Adv. Math., 158(2) p. 154-168 (2001). Zbl1193.57002MR1822682
  12. Pilgrim (K.).— An algebraic formulation of Thurston’s characterization of rational functions, Ann. Fac. Sci. Toulouse Math (6), 21(5), (2012). Zbl1272.37025
  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). Zbl0774.58035MR1149864
  14. Selinger (N.).— Thurston’s pullback map on the augmented Teichmüller space and applications. arXiv:1010.1690v2. Zbl1298.37033
  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). Zbl1062.37039MR1765095
  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). Zbl0969.37020MR1758798

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