Nevanlinna theoretical exceptional sets of rational towers and semigroups

Okuyama, Yûsuke

doi:10.1090/S1088-4173-06-00140-8

Nevanlinna theoretical exceptional sets of rational towers and semigroups
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by Yûsuke Okuyama

Conform. Geom. Dyn. 10 (2006), 100-116

DOI: https://doi.org/10.1090/S1088-4173-06-00140-8

Published electronically: April 6, 2006

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Abstract:

For a rational tower, i.e., a composition sequence of rational maps, in addition to the algebraic and dynamical exceptional sets, various Nevanlinna theoretical exceptional sets are defined, and as we showed previously in the case of iterations, all of them are the same. In this paper, we extend this result to the cases of a rational tower with summable distortions and a finitely generated rational semigroup. We show that all the exceptional sets of a finitely generated rational semigroup are countable, and all of them are empty if and only if the algebraic one is as well (this being the smallest among them). The countability of exceptional sets is fundamental in the Nevanlinna theory, and their emptiness is important in the complex dynamics.

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