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On the Bieberbach conjecture and holomorphic dynamics

Author: Xavier Buff
Journal: Proc. Amer. Math. Soc. 131 (2003), 755-759
MSC (2000): Primary 37F10, 30C50
Published electronically: October 18, 2002
MathSciNet review: 1937413
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Abstract: In this note we prove that when $P$ is a polynomial of degree $d$ with connected Julia set and when $z_0$ belongs to the filled-in Julia set $K(P)$, then $\vert P'(z_0)\vert\leq d^2$. We also show that equality is achieved if and only if $K(P)$ is a segment of which one extremity is $z_0$. In that case, $P$ is conjugate to a Tchebycheff polynomial or its opposite. The main tool in our proof is the Bieberbach conjecture proved by de Branges in 1984.

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  • [A] Lars V. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429. MR 0167618
  • [B] Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. MR 0297992
  • [H] 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
  • [L] Kin Y. Li, Interpolating Blaschke products and the left spectrum of multiplication operators on the Bergman space, Hokkaido Math. J. 21 (1992), no. 2, 295–304. MR 1169796, 10.14492/hokmj/1381413684
  • [McM] C. McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990), no. 2, 425–454. MR 1031909, 10.1007/BF01234427
  • [M] Peter Raith, Perturbations of a topologically transitive piecewise monotonic map on the interval, Iteration theory (ECIT ’96) (Urbino), Grazer Math. Ber., vol. 339, Karl-Franzens-Univ. Graz, Graz, 1999, pp. 301–312. MR 1748832
  • [O] Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996), x+159 (French, with French summary). MR 1402300
  • [Pe] Carsten Lunde Petersen, On the Pommerenke-Levin-Yoccoz inequality, Ergodic Theory Dynam. Systems 13 (1993), no. 4, 785–806. MR 1257034
  • [Po] Ch. Pommerenke, On conformal mapping and iteration of rational functions, Complex Variables Theory Appl. 5 (1986), no. 2-4, 117–126. MR 846481
  • [Y] J.C. YOCCOZ. Sur la taille des membres de l'Ensemble de Mandelbrot. (1987) (unpublished).

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Additional Information

Xavier Buff
Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France

Received by editor(s): June 25, 2001
Received by editor(s) in revised form: August 14, 2001
Published electronically: October 18, 2002
Communicated by: Linda Keen
Article copyright: © Copyright 2002 American Mathematical Society