A parabolic Pommerenke-Levin-Yoccoz inequality

• Autorzy: Xavier Buff , Adam L. Epstein
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
• 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
• 37F10: Polynomials; rational maps; entire and meromorphic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2002
• Tom: 172
• Numer: 3
• Strony: 249-289

Daty

wydano

2002

Twórcy

autor

Xavier Buff

• Laboratoire Émile Picard, Université Paul Sabatier, 31062 Toulouse Cedex, France

autor

Adam L. Epstein

• Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Identyfikatory

DOI

10.4064/fm172-3-3