Quasi-conformal rigidity of multicritical maps

Peng, Wenjuan; Tan, Lei

doi:10.1090/S0002-9947-2014-06140-0

Quasi-conformal rigidity of multicritical maps
HTML articles powered by AMS MathViewer

by Wenjuan Peng and Lei Tan PDF

Trans. Amer. Math. Soc. 367 (2015), 1151-1182 Request permission

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.

References

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, DOI 10.1090/ulect/038
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, DOI 10.4007/annals.2009.170.783
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, DOI 10.1007/BF02392761
GuiZhen Cui and WenJuan Peng, On the structure of Fatou domains, Sci. China Ser. A 51 (2008), no. 7, 1167–1186. MR 2417486, DOI 10.1007/s11425-008-0056-5
Guizhen Cui and Lei Tan, A characterization of hyperbolic rational maps, Invent. Math. 183 (2011), no. 3, 451–516. MR 2772086, DOI 10.1007/s00222-010-0281-8
A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, II., Publ. Math. dOrsay, 1984–1985.
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
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, DOI 10.1007/BF02392534
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
H. Inou and M. Shishikura, The renormalization for parabolic fixed points and their perturbation, preprint.
J. Kahn, A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics, arXiv:math/0609045.
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, DOI 10.24033/asens.2063
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, DOI 10.1201/b10617-7
Jeremy Kahn and Mikahil Lyubich, The quasi-additivity law in conformal geometry, Ann. of Math. (2) 169 (2009), no. 2, 561–593. MR 2480612, DOI 10.4007/annals.2009.169.561
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, DOI 10.4007/annals.2009.170.413
O. Kozlovski, W. Shen, and S. van Strien, Rigidity for real polynomials, Ann. of Math. (2) 165 (2007), no. 3, 749–841. MR 2335796, DOI 10.4007/annals.2007.165.749
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, DOI 10.1112/plms/pdn055
Genadi Levin, Multipliers of periodic orbits of quadratic polynomials and the parameter plane, Israel J. Math. 170 (2009), 285–315. MR 2506328, DOI 10.1007/s11856-009-0030-0
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, DOI 10.1007/s00220-011-1228-7
Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
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
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
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, DOI 10.1090/S1088-4173-10-00201-8
WenJuan Peng and Lei Tan, Combinatorial rigidity of unicritical maps, Sci. China Math. 53 (2010), no. 3, 831–848. MR 2608337, DOI 10.1007/s11425-010-0022-x
Feliks Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fund. Math. 149 (1996), no. 2, 95–118. MR 1376666, DOI 10.4064/fm-149-2-95-118
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, DOI 10.1007/s11425-008-0178-9
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, DOI 10.4007/annals.2008.168.127
M. Shishikura, Yoccoz puzzle, $\tau$-functions and their applications, Unpublished, 1998.
Lei Tan and Yongcheng Yin, The unicritical Branner-Hubbard conjecture, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 215–227. MR 2508258
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, DOI 10.1201/b10617-3
J. -C. Yoccoz, On the local connectivity of the Mandelbrot sets, Unpublished, 1990.
Yongcheng Yin and Yu Zhai, No invariant line fields on Cantor Julia sets, Forum Math. 22 (2010), no. 1, 75–94. MR 2604364, DOI 10.1515/FORUM.2010.004
Yu Zhai, Rigidity for rational maps with Cantor Julia sets, Sci. China Ser. A 51 (2008), no. 1, 79–92. MR 2390757, DOI 10.1007/s11425-007-0124-2