In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. 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 ℜ((N+1)/2 - ι(f,β)) > N/π².
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-3-3,
author = {Xavier Buff and Adam L. Epstein},
title = {A parabolic Pommerenke-Levin-Yoccoz inequality},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {249-289},
zbl = {1115.37323},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-3-3}
}
Xavier Buff; Adam L. Epstein. A parabolic Pommerenke-Levin-Yoccoz inequality. Fundamenta Mathematicae, Tome 173 (2002) pp. 249-289. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm172-3-3/