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On the Pommerenke-Levin-Yoccoz inequality. (English) Zbl 0802.30022
Let \(R\) be a rational function viewed as a self-mapping of the Riemann sphere and \(\alpha\) be a repelling or parabolic periodic point of \(R\) on the boundary of a simply connected Fatou domain \(\Lambda\). The author establishes a sufficient criterion for the existence of a periodic access to \(\alpha\) in \(\Lambda\). To this end, the author defines a (combinatorial) rotation number for \(\alpha\). These results generalize theorems due to Douady, Eremenko and Levin about landing external rays for polynomials. Let \(B'\) be the Blaschke product corresponding to \(R |_ \Lambda\) and \(\alpha'\) be the repelling periodic point of \(B'\) corresponding to \(\alpha\). In order to describe relations between the multipliers of \(\alpha\) and \(\alpha'\) the author proves an inequality merging earlier results obtained by Pommerenke, Levin and Yoccoz. The main ingredients of the proofs are linearizing maps and Grötzsch inequalities for annuli in a torus.
Reviewer: H.Kriete (Aachen)

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics