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A tableau approach of the KSS nest


Authors: Wenjuan Peng, Weiyuan Qiu, Pascale Roesch, Lei Tan and Yongcheng Yin
Journal: Conform. Geom. Dyn. 14 (2010), 35-67
MSC (2010): Primary 32H50, 37F10, 37F20
DOI: https://doi.org/10.1090/S1088-4173-10-00201-8
Published electronically: February 18, 2010
MathSciNet review: 2600535
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Abstract | References | Similar Articles | Additional Information

Abstract: The KSS nest is a sophisticated choice of puzzle pieces given in [Ann. of Math. 165 (2007), 749-841]. This nest, once combined with the KL-Lemma, has proven to be a powerful machinery, leading to several important advancements in the field of holomorphic dynamics. We give here a presentation of the KSS nest in terms of tableau. This is an effective language invented by Branner and Hubbard to deal with the complexity of the dynamics of puzzle pieces. We show, in a typical situation, how to make the combination between the KSS nest and the KL-Lemma. One consequence of this is the recently proved Branner-Hubbard conjecture. Our estimates here can be used to give an alternative proof of the rigidity property.


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  • [AKLS] A. Avila, J. Kahn, M. Lyubich and W. Shen, Combinatorial rigidity for unicritical polynomials, ArXiv: math.DS/0507240. Annals of Math, Sept. 2009.
  • [BH] B. Branner and J. H. Hubbard, The iteration of cubic polynomials, Part II: Patterns and parapatterns, Acta Math., 169 (1992), 229-325. MR 1194004 (94d:30044)
  • [KL1] J. Kahn and M. Lyubich, The quasi-additivity law in conformal geometry, Annals of Math. 169 (2009), No. 2, 561-593. MR 2480612
  • [KL2] J. Kahn and M. Lyubich, Local connectivity of Julia sets for unicritical polynomials, ArXiv: math.DS/0505194, Stony Brook IMS preprint 2005/3. Annals of Math, Sept. 2009.
  • [KSS] O. Kozlovski, W. Shen, and S. van Strien, Rigidity for real polynomials, Annals of Mathematics 165 (2007), 749-841. MR 2335796 (2008m:37063)
  • [KS] O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, ArXiv: math.DS/0609710. Accepted for publication in the Proceedings of the LMS.
  • [M1] J. Milnor, Local connectivity of Julia sets: expository lectures. In: The Mandelbrot set, Theme and Variations, edited by Tan Lei, London Math. Soc. Lecture Note Ser., No 274, Cambridge Univ. Press, 2000, 67-116. MR 1765085 (2001b:37073)
  • [M2] John Milnor, Dynamics in One Variable, Annals of Mathematics Studies, 160, Princeton University, 2006. MR 2193309 (2006g:37070)
  • [QY] Qiu Weiyuan and Yin Yongcheng, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Science in China, Series A, 2009, Vol. 52, No. 1, 45-65. MR 2471515 (2009j:37074)
  • [PT] Peng Wenjuan and Tan Lei, Combinatorial rigidity of unicritical maps, to appear in: Science in China.
  • [Ro] Pascale Roesch, Puzzles de Yoccoz pour les applications à allure rationnelle, L'Enseignement Mathématique, tome 45, Juin 1999, pp. 133-168. MR 1703365 (2000g:37050)
  • [RY] Pascale Roesch and Yin Yongcheng, The boundary of bounded polynomial Fatou components, C. R. Acad. Sci. Paris, Ser. I 346 (2008). MR 2441925
  • [TY] Tan Lei and Yin Yongcheng, Unicritical Branner-Hubbard conjecture, in Family and friends, ed. Dierk Schleicher.
  • [YZ] Yin Yongcheng and Zhai Yu, No invariant line fields on Cantor Julia sets, to appear in: Forum Mathematicum.

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Additional Information

Wenjuan Peng
Affiliation: School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Email: wenjpeng@amss.ac.cn

Weiyuan Qiu
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China
Email: wyqiu@fudan.edu.cn

Pascale Roesch
Affiliation: Laboratoire Émile-Picard, Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 9, France
Email: roesch@math.univ-toulouse.fr

Lei Tan
Affiliation: Université d’Angers, Faculté des Sciences, LAREMA, 2, Boulevard Lavoisier, 49045 Angers cedex 01, France
Email: Lei.Tan@univ-angers.fr

Yongcheng Yin
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China
Email: ycyin@fudan.edu.cn

DOI: https://doi.org/10.1090/S1088-4173-10-00201-8
Received by editor(s): March 6, 2009
Published electronically: February 18, 2010
Additional Notes: The first author is supported by China Postdoctoral Science Foundation under Grant No. 20080440270, National Natural Science Foundation of China under Grant No. 10831004 and the Doctoral Education Program Foundation of China under Grant No. 20060001003
The second author is supported by National Natural Science Foundation of China under Grants No. 10831004 and 10871047
The third author is supported by EU Research Training Network CODY, Conformal Structures and Dynamics
The fourth author is supported by National Natural Science Foundation of China under Grant No. 10831004
The fifth author is supported by the project ABC of the Agence Nationale de la Recherche Francaise
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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