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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)

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