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Conformal Geometry and Dynamics

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Nevanlinna theoretical exceptional sets of rational towers and semigroups

Author: Yûsuke Okuyama
Journal: Conform. Geom. Dyn. 10 (2006), 100-116
MSC (2000): Primary 30D35; Secondary 37F15, 32H50
Published electronically: April 6, 2006
MathSciNet review: 2218642
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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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Additional Information

Yûsuke Okuyama
Affiliation: Department of Mathematics, Faculty of Science, Kanazawa University, Kanazawa 920-1192 Japan
Address at time of publication: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin katu 2b), FI-00014 Helsinki, Finland

Keywords: Nevanlinna theory, complex dynamics, rational tower, rational semigroup, algebraic exceptional set, Picard exceptional set, dynamical exceptional set, Nevanlinna exceptional set, Valiron exceptional set
Received by editor(s): April 19, 2005
Received by editor(s) in revised form: October 13, 2005
Published electronically: April 6, 2006
Additional Notes: Partially supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan, Grant-in-Aid for Young Scientists (B), 15740085, 2004
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.