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Conformal Geometry and Dynamics

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Combinatorial rigidity for some infinitely renormalizable unicritical polynomials


Author: Davoud Cheraghi
Journal: Conform. Geom. Dyn. 14 (2010), 219-255
MSC (2010): Primary 37F45; Secondary 37F25, 37F30
DOI: https://doi.org/10.1090/S1088-4173-2010-00216-X
Published electronically: September 15, 2010
MathSciNet review: 2719786
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Abstract: Here we prove that infinitely renormalizable unicritical polynomials $ P_c:z \mapsto z^d+c$, with $ c\in\mathbb{C}$, satisfying a priori bounds and a certain ``combinatorial'' condition are combinatorially rigid. This implies the local connectivity of the connectedness loci (the Mandelbrot set when $ d=2$) at the corresponding parameters.


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  • [Ahl06] L. V. Ahlfors, Lectures on quasiconformal mappings, second ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. MR 2241787 (2009d:30001)
  • [AKLS09] A. Avila, J. Kahn, M. Lyubich, and Weixiao Shen, Combinatorial rigidity for unicritical polynomials, Ann. of Math. (2) 170 (2009), no. 2, 783-797. MR 2552107
  • [Bra94] B. Branner, Puzzles and para-puzzles of quadratic and cubic polynomials, Complex dynamical systems (Cincinnati, OH, 1994), Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 31-69. MR 1315533
  • [DH85a] A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, II, Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], vol. 84-85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984-1985.
  • [DH85b] -, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287-343. MR 816367 (87f:58083)
  • [Dou93] A. Douady, Descriptions of compact sets in $ {\bf C}$, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 429-465. MR 1215973 (94g:58185)
  • [GŚ98] J. Graczyk and G. Świa tek, The real Fatou conjecture, Annals of Mathematics Studies, vol. 144, Princeton University Press, Princeton, NJ, 1998.
  • [Hub93] 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)
  • [Jia00] Y. Jiang, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5077-5091. MR 1675198 (2001b:37071)
  • [Kah06] J. Kahn, A priori bounds for some infinitely renormalizable maps: I. bounded primitive combinatorics, Preprint, IMS at Stony Brook, 2006/05.
  • [KL08] J. Kahn and M. Lyubich, A priori bounds for some infinitely renormalizable quadratics. II. Decorations, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 1, 57-84. MR 2423310 (2009k:37106)
  • [KL09a] -, Local connectivity of Julia sets for unicritical polynomials, Ann. of Math. (2) 170 (2009), no. 1, 413-426. MR 2521120 (2010h:37094)
  • [KL09b] -, 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)
  • [KSvS07] 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)
  • [Lev09] G. Levin, Multipliers of periodic orbits of quadratic polynomials and the parameter plane, Israel J. Math. 170 (2009), 285-315. MR 2506328
  • [LvS98] G. Levin and S. van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. (2) 147 (1998), no. 3, 471-541. MR 1637647 (99e:58143)
  • [Lyu97] M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta. Math. 178 (1997), 185-247, 247-297. MR 1459261 (98e:58145)
  • [McM94] C. T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365 (96b:58097)
  • [Mil00a] J. 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)
  • [Mil00b] -, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Astérisque (2000), no. 261, xiii, 277-333, Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755445 (2002e:37067)
  • [Mil06] -, Dynamics in one complex variable, third ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309 (2006g:37070)
  • [Sch04] D. Schleicher, On fibers and local connectivity of Mandelbrot and Multibrot sets, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 477-517. MR 2112117 (2006b:37088)
  • [Slo91] Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347-355. MR 1037218 (91f:58078)
  • [Str55] K. Strebel, On the maximal dilation of quasiconformal mappings, Proc. Amer. Math. Soc. 6 (1955), 903-909. MR 0073702 (17:473d)
  • [Sul92] D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), Amer. Math. Soc., Providence, RI, 1992, pp. 417-466. MR 1184622 (93k:58194)

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Additional Information

Davoud Cheraghi
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Address at time of publication: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: d.cheraghi@warwick.ac.uk

DOI: https://doi.org/10.1090/S1088-4173-2010-00216-X
Received by editor(s): April 20, 2008
Received by editor(s) in revised form: December 28, 2009, and June 2, 2010
Published electronically: September 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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