Combinatorial rigidity for some infinitely renormalizable unicritical polynomials

Cheraghi, Davoud

doi:10.1090/S1088-4173-2010-00216-X

Combinatorial rigidity for some infinitely renormalizable unicritical polynomials
HTML articles powered by AMS MathViewer

by Davoud Cheraghi

Conform. Geom. Dyn. 14 (2010), 219-255

DOI: https://doi.org/10.1090/S1088-4173-2010-00216-X

Published electronically: September 15, 2010

PDF | Request permission

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.

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, 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, DOI 10.1090/psapm/049/1315533
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.
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, DOI 10.24033/asens.1491
Adrien Douady, Descriptions of compact sets in $\textbf {C}$, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429–465. MR 1215973
J. Graczyk and G. Świa̧tek, The real Fatou conjecture, Annals of Mathematics Studies, vol. 144, Princeton University Press, Princeton, NJ, 1998.
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
Yunping Jiang, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5077–5091. MR 1675198, DOI 10.1090/S0002-9947-00-02514-9
J. Kahn, A priori bounds for some infinitely renormalizable maps: I. bounded primitive combinatorics, Preprint, IMS at Stony Brook, 2006/05.
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, 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
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
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
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 and Sebastian van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. (2) 147 (1998), no. 3, 471–541. MR 1637647, DOI 10.2307/120958
Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
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, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Astérisque 261 (2000), xiii, 277–333 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755445
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
Dierk 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, DOI 10.1090/pspum/072.1/2112117
Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347–355. MR 1037218, DOI 10.1090/S0002-9939-1991-1037218-8
Kurt Strebel, On the maximal dilation of quasiconformal mappings, Proc. Amer. Math. Soc. 6 (1955), 903–909. MR 73702, DOI 10.1090/S0002-9939-1955-0073702-X
Dennis 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