A parabolic Pommerenke–Levin–Yoccoz inequality

Volume 172 / 2002

Xavier Buff, Adam L. Epstein Fundamenta Mathematicae 172 (2002), 249-289 MSC: Primary 37F10; Secondary 30D05, 37F45. DOI: 10.4064/fm172-3-3


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 $$ \Re ((N+1)/{2}-\iota (f,\beta )) > N/{\pi ^2}.$$


  • Xavier BuffLaboratoire Émile Picard
    Université Paul Sabatier
    31062 Toulouse Cedex, France
  • Adam L. EpsteinMathematics Institute
    University of Warwick
    Coventry CV4 7AL, United Kingdom

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