## On the Bieberbach conjecture and holomorphic dynamics

HTML articles powered by AMS MathViewer

- by Xavier Buff PDF
- Proc. Amer. Math. Soc.
**131**(2003), 755-759 Request permission

## 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 $|P’(z_0)|\leq d^2$. 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.## References

- Lars V. Ahlfors,
*Finitely generated Kleinian groups*, Amer. J. Math.**86**(1964), 413–429. MR**167618**, DOI 10.2307/2373173 - Lipman Bers,
*On boundaries of Teichmüller spaces and on Kleinian groups. I*, Ann. of Math. (2)**91**(1970), 570–600. MR**297992**, DOI 10.2307/1970638 - 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** - 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**, DOI 10.14492/hokmj/1381413684 - C. McMullen,
*Iteration on Teichmüller space*, Invent. Math.**99**(1990), no. 2, 425–454. MR**1031909**, DOI 10.1007/BF01234427 - 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** - 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** - Carsten Lunde Petersen,
*On the Pommerenke-Levin-Yoccoz inequality*, Ergodic Theory Dynam. Systems**13**(1993), no. 4, 785–806. MR**1257034** - Ch. Pommerenke,
*On conformal mapping and iteration of rational functions*, Complex Variables Theory Appl.**5**(1986), no. 2-4, 117–126. MR**846481**, DOI 10.1080/17476938608814133 - J.C. Yoccoz.
*Sur la taille des membres de l’Ensemble de Mandelbrot.*(1987) (unpublished).

## Additional Information

**Xavier Buff**- Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France
- Email: buff@picard.ups-tlse.fr
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 755-759 - MSC (2000): Primary 37F10, 30C50
- DOI: https://doi.org/10.1090/S0002-9939-02-06864-8
- MathSciNet review: 1937413