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Quasi-conformal rigidity of multicritical maps


Authors: Wenjuan Peng and Lei Tan
Journal: Trans. Amer. Math. Soc. 367 (2015), 1151-1182
MSC (2010): Primary 37F10, 37F20
DOI: https://doi.org/10.1090/S0002-9947-2014-06140-0
Published electronically: July 25, 2014
MathSciNet review: 3280040
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Abstract | References | Similar Articles | Additional Information

Abstract: We combine the enhanced nest constructed by Kozlovski, Shen and van Strien, and the analytic method proposed by Avila, Kahn, Lyubich and Shen to prove quasi-conformal rigidity properties of multicritical maps.


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  • [1] Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787 (2009d:30001)
  • [2] Artur Avila, Jeremy Kahn, Mikhail Lyubich, and Weixiao Shen, Combinatorial rigidity for unicritical polynomials, Ann. of Math. (2) 170 (2009), no. 2, 783-797. MR 2552107 (2011d:37078), https://doi.org/10.4007/annals.2009.170.783
  • [3] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229-325. MR 1194004 (94d:30044), https://doi.org/10.1007/BF02392761
  • [4] GuiZhen Cui and WenJuan Peng, On the structure of Fatou domains, Sci. China Ser. A 51 (2008), no. 7, 1167-1186. MR 2417486 (2009e:30050), https://doi.org/10.1007/s11425-008-0056-5
  • [5] Guizhen Cui and Lei Tan, A characterization of hyperbolic rational maps, Invent. Math. 183 (2011), no. 3, 451-516. MR 2772086 (2012c:37088), https://doi.org/10.1007/s00222-010-0281-8
  • [6] A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, II., Publ. Math. dOrsay, 1984-1985.
  • [7] Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287-343. MR 816367 (87f:58083)
  • [8] Adrien Douady and John H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263-297. MR 1251582 (94j:58143), https://doi.org/10.1007/BF02392534
  • [9] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467-511. MR 1215974 (94c:58172)
  • [10] H. Inou and M. Shishikura, The renormalization for parabolic fixed points and their perturbation, preprint.
  • [11] J. Kahn, A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics, arXiv:math/0609045.
  • [12] Jeremy Kahn and Mikhail Lyubich, A priori bounds for some infinitely renormalizable quadratics. II. Decorations, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 57-84 (English, with English and French summaries). MR 2423310 (2009k:37106)
  • [13] Jeremy Kahn and Mikhail Lyubich, A priori bounds for some infinitely renormalizable quadratics. III. Molecules, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 229-254. MR 2508259 (2010f:37078), https://doi.org/10.1201/b10617-7
  • [14] Jeremy Kahn and Mikahil Lyubich, The quasi-additivity law in conformal geometry, Ann. of Math. (2) 169 (2009), no. 2, 561-593. MR 2480612 (2010a:37091), https://doi.org/10.4007/annals.2009.169.561
  • [15] Jeremy Kahn and Mikhail Lyubich, Local connectivity of Julia sets for unicritical polynomials, Ann. of Math. (2) 170 (2009), no. 1, 413-426. MR 2521120 (2010h:37094), https://doi.org/10.4007/annals.2009.170.413
  • [16] O. Kozlovski, W. Shen, and S. van Strien, Rigidity for real polynomials, Ann. of Math. (2) 165 (2007), no. 3, 749-841. MR 2335796 (2008m:37063), https://doi.org/10.4007/annals.2007.165.749
  • [17] Oleg Kozlovski and Sebastian van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3) 99 (2009), no. 2, 275-296. MR 2533666 (2011a:37096), https://doi.org/10.1112/plms/pdn055
  • [18] Genadi Levin, Multipliers of periodic orbits of quadratic polynomials and the parameter plane, Israel J. Math. 170 (2009), 285-315. MR 2506328 (2011d:37082), https://doi.org/10.1007/s11856-009-0030-0
  • [19] Genadi Levin, Rigidity and non-local connectivity of Julia sets of some quadratic polynomials, Comm. Math. Phys. 304 (2011), no. 2, 295-328. MR 2795323 (2012c:37089), https://doi.org/10.1007/s00220-011-1228-7
  • [20] Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185-247, 247-297. MR 1459261 (98e:58145), https://doi.org/10.1007/BF02392694
  • [21] John Milnor, Local connectivity of Julia sets: expository lectures, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 67-116. MR 1765085 (2001b:37073)
  • [22] John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • [23] Wenjuan Peng, Weiyuan Qiu, Pascale Roesch, Lei Tan, and Yongcheng Yin, A tableau approach of the KSS nest, Conform. Geom. Dyn. 14 (2010), 35-67. MR 2600535 (2011d:37079), https://doi.org/10.1090/S1088-4173-10-00201-8
  • [24] WenJuan Peng and Lei Tan, Combinatorial rigidity of unicritical maps, Sci. China Math. 53 (2010), no. 3, 831-848. MR 2608337 (2011i:37060), https://doi.org/10.1007/s11425-010-0022-x
  • [25] Feliks Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fund. Math. 149 (1996), no. 2, 95-118. MR 1376666 (97e:58199)
  • [26] WeiYuan Qiu and YongCheng Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Sci. China Ser. A 52 (2009), no. 1, 45-65. MR 2471515 (2009j:37074), https://doi.org/10.1007/s11425-008-0178-9
  • [27] P. Roesch, On local connectivity for the Julia set of rational maps: Newton's famous example, Ann. of Math. (2) 168 (2008), no. 1, 127-174. MR 2415400 (2009c:37043), https://doi.org/10.4007/annals.2008.168.127
  • [28] M. Shishikura, Yoccoz puzzle, $ \tau $-functions and their applications, Unpublished, 1998.
  • [29] Lei Tan and Yongcheng Yin, The unicritical Branner-Hubbard conjecture, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 215-227. MR 2508258 (2010f:37077)
  • [30] William P. Thurston, On the geometry and dynamics of iterated rational maps, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 3-137. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. MR 2508255 (2010m:37076), https://doi.org/10.1201/b10617-3
  • [31] J. -C. Yoccoz, On the local connectivity of the Mandelbrot sets, Unpublished, 1990.
  • [32] Yongcheng Yin and Yu Zhai, No invariant line fields on Cantor Julia sets, Forum Math. 22 (2010), no. 1, 75-94. MR 2604364 (2011d:37077), https://doi.org/10.1515/FORUM.2010.004
  • [33] Yu Zhai, Rigidity for rational maps with Cantor Julia sets, Sci. China Ser. A 51 (2008), no. 1, 79-92. MR 2390757 (2009i:37110), https://doi.org/10.1007/s11425-007-0124-2

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Additional Information

Wenjuan Peng
Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, People’s Republic of China
Email: wenjpeng@amss.ac.cn

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

DOI: https://doi.org/10.1090/S0002-9947-2014-06140-0
Keywords: Holomorphic dynamics, quasi-conformal rigidity, multicritical maps, enhanced nest
Received by editor(s): January 20, 2012
Received by editor(s) in revised form: January 30, 2013
Published electronically: July 25, 2014
Additional Notes: The first author was supported by the NSF of China under grants No. 11101402 and No. 11231009, by the PSSF of China under grant No. 201003020 and by SRF for ROCS, SEM
The second author was supported by Geanpyl Pays de la Loire and ANR LAMBDA
Article copyright: © Copyright 2014 American Mathematical Society

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