A parabolic Pommerenke–Levin–Yoccoz inequality

Xavier Buff¹ ; Adam L. Epstein²

¹ Laboratoire Émile Picard Université Paul Sabatier 31062 Toulouse Cedex, France
² Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom

Fundamenta Mathematicae, Tome 172 (2002) no. 3, pp. 249-289.

Résumé

In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point $\beta $ of a rational map $f: {\mathbb P}^1\to {\mathbb P}^1$, the number ${N}$ of attracting petals and the residue $\iota (f,\beta )$ of the 1-form $dz/(z-f(z))$ at $\beta $. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only ${N}$ grand orbit equivalence classes of critical points in the immediate basin, then $$ \Re ((N+1)/{2}-\iota (f,\beta )) > N/{\pi ^2}.$$

Détail

DOI : 10.4064/fm172-3-3

Keywords: recent preprint bergweiler relates number critical points contained immediate basin multiple fixed point beta rational map mathbb mathbb number attracting petals residue iota beta form z f beta article present different approach problem which developing independently time apply method answer question raised bergweiler particular prove there only grand orbit equivalence classes critical points immediate basin iota beta