## 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.## References

- Kazuo Azukawa,
*Hyperbolicity of circular domains*, Tohoku Math. J. (2)**35**(1983), no. 3, 403–413. MR**711356**, DOI 10.2748/tmj/1178228999 - Hans Brolin,
*Invariant sets under iteration of rational functions*, Ark. Mat.**6**(1965), 103–144 (1965). MR**194595**, DOI 10.1007/BF02591353 - Laura DeMarco,
*Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity*, Math. Ann.**326**(2003), no. 1, 43–73. MR**1981611**, DOI 10.1007/s00208-002-0404-7 - Tien-Cuong Dinh,
*Suites d’applications méromorphes multivaluées et courants laminaires*, J. Geom. Anal.**15**(2005), no. 2, 207–227 (French, with English summary). MR**2152480**, DOI 10.1007/BF02922193
DS03 T.-C. Dinh and N. Sibony, Value distribution of meromorphic transforms and applications, math.DS/0306095.
- A. È. Erëmenko and M. L. Sodin,
*Iterations of rational functions and the distribution of the values of Poincaré functions*, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen.**53**(1990), 18–25 (Russian); English transl., J. Soviet Math.**58**(1992), no. 6, 504–509. MR**1077218**, DOI 10.1007/BF01109688 - Herbert Federer,
*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR**0257325** - John Erik Fornaess and Nessim Sibony,
*Complex dynamics in higher dimension. II*, Modern methods in complex analysis (Princeton, NJ, 1992) Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 135–182. MR**1369137** - Ricardo Mañé,
*On the uniqueness of the maximizing measure for rational maps*, Bol. Soc. Brasil. Mat.**14**(1983), no. 1, 27–43. MR**736567**, DOI 10.1007/BF02584743 - Aimo Hinkkanen,
*Sharp error term in the Nevanlinna theory*, Complex differential and functional equations (Mekrijärvi, 2000) Univ. Joensuu Dept. Math. Rep. Ser., vol. 5, Univ. Joensuu, Joensuu, 2003, pp. 51–79. MR**1968110** - A. Hinkkanen and G. J. Martin,
*Some properties of semigroups of rational functions*, XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995) de Gruyter, Berlin, 1996, pp. 53–58. MR**1427070** - John H. Hubbard and Peter Papadopol,
*Superattractive fixed points in $\textbf {C}^n$*, Indiana Univ. Math. J.**43**(1994), no. 1, 321–365. MR**1275463**, DOI 10.1512/iumj.1994.43.43014 - Mattias Jonsson,
*Ergodic properties of fibered rational maps*, Ark. Mat.**38**(2000), no. 2, 281–317. MR**1785403**, DOI 10.1007/BF02384321 - Maciej Klimek,
*Pluripotential theory*, London Mathematical Society Monographs. New Series, vol. 6, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR**1150978** - Ilpo Laine,
*Nevanlinna theory and complex differential equations*, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR**1207139**, DOI 10.1515/9783110863147 - M. Ju. Ljubich,
*Entropy properties of rational endomorphisms of the Riemann sphere*, Ergodic Theory Dynam. Systems**3**(1983), no. 3, 351–385. MR**741393**, DOI 10.1017/S0143385700002030 - Ricardo Mañé,
*The Hausdorff dimension of invariant probabilities of rational maps*, Dynamical systems, Valparaiso 1986, Lecture Notes in Math., vol. 1331, Springer, Berlin, 1988, pp. 86–117. MR**961095**, DOI 10.1007/BFb0083068 - S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda,
*Holomorphic dynamics*, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000. Translated from the 1995 Japanese original and revised by the authors. MR**1747010** - Junjiro Noguchi and Takushiro Ochiai,
*Geometric function theory in several complex variables*, Translations of Mathematical Monographs, vol. 80, American Mathematical Society, Providence, RI, 1990. Translated from the Japanese by Noguchi. MR**1084378**, DOI 10.1090/mmono/080 - Yûsuke Okuyama,
*Nevanlinna, Siegel, and Cremer*, Indiana Univ. Math. J.**53**(2004), no. 3, 755–763. MR**2086699**, DOI 10.1512/iumj.2004.53.2503 - Yûsuke Okuyama,
*Complex dynamics, value distributions, and potential theory*, Ann. Acad. Sci. Fenn. Math.**30**(2005), no. 2, 303–311. MR**2173366** - Yûsuke Okuyama,
*Valiron, Nevanlinna and Picard exceptional sets of iterations of rational functions*, Proc. Japan Acad. Ser. A Math. Sci.**81**(2005), no. 2, 23–26. MR**2126072** - Bernard Shiffman and Steve Zelditch,
*Distribution of zeros of random and quantum chaotic sections of positive line bundles*, Comm. Math. Phys.**200**(1999), no. 3, 661–683. MR**1675133**, DOI 10.1007/s002200050544 - M. Sodin,
*Value distribution of sequences of rational functions*, Entire and subharmonic functions, Adv. Soviet Math., vol. 11, Amer. Math. Soc., Providence, RI, 1992, pp. 7–20. MR**1188001** - Hiroki Sumi,
*Skew product maps related to finitely generated rational semigroups*, Nonlinearity**13**(2000), no. 4, 995–1019. MR**1767945**, DOI 10.1088/0951-7715/13/4/302
SumiLyubich H. Sumi, unpublished manuscript (2001).
Tortrat87 P. Tortrat, Aspects potentialistes de l’itération des polynômes, Séminaire de Théorie du Potentiel, Paris, No. 8, Vol. 1235 of - Tetsuo Ueda,
*Fatou sets in complex dynamics on projective spaces*, J. Math. Soc. Japan**46**(1994), no. 3, 545–555. MR**1276837**, DOI 10.2969/jmsj/04630545

*Lecture Notes in Math.*, Springer, Berlin (1987), 195–209.

## Bibliographic 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
- Email: yusuke.okuyama@helsinki.fi
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**10**(2006), 100-116 - MSC (2000): Primary 30D35; Secondary 37F15, 32H50
- DOI: https://doi.org/10.1090/S1088-4173-06-00140-8
- MathSciNet review: 2218642