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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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  • 1. K. Azukawa, Hyperbolicity of circular domains, Tohoku Math. J. (2) 35, 3 (1983), 403-413. MR 0711356 (85c:32041)
  • 2. H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. MR 0194595 (33:2805)
  • 3. L. DeMarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326, 1 (2003), 43-73. MR 1981611 (2004f:32044)
  • 4. T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires, J. Geom. Anal. 15, 2 (2005), 207-227. MR 2152480
  • 5. T.-C. Dinh and N. Sibony, Value distribution of meromorphic transforms and applications, math.DS/0306095.
  • 6. A. E. Erëmenko and M. L. Sodin, Iterations of rational functions and the distribution of the values of Poincaré functions, Teor. Funktsi{\u{\i\/}}\kern.15emFunktsional. Anal. i Prilozhen. 53 (1990), 18-25. MR 1077218 (92d:30016)
  • 7. H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York (1969). MR 0257325 (41:1976)
  • 8. J. E. Fornaess and N. Sibony, Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992), Vol. 137 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ (1995), 135-182. MR 1369137 (97g:32033)
  • 9. A. Freire, A. Lopes, and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14, 1 (1983), 45-62. MR 0736568 (85m:58110b)
  • 10. A. Hinkkanen, Sharp error term in the Nevanlinna theory, Complex differential and functional equations (Mekrijärvi, 2000), Vol. 5 of Univ. Joensuu Dept. Math. Rep. Ser., Univ. Joensuu, Joensuu (2003), 51-79. MR 1968110 (2004i:30021)
  • 11. A. Hinkkanen and G. J. Martin, Some properties of semigroups of rational functions, XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995), de Gruyter, Berlin (1996), 53-58. MR 1427070 (97i:30036)
  • 12. J. H. Hubbard and P. Papadopol, Superattractive fixed points in $ {\bf C}\sp n$, Indiana Univ. Math. J. 43, 1 (1994), 321-365. MR 1275463 (95e:32025)
  • 13. M. Jonsson, Ergodic properties of fibered rational maps, Ark. Mat. 38, 2 (2000), 281-317. MR 1785403 (2002k:37073)
  • 14. M. Klimek, Pluripotential theory, Vol. 6 of London Mathematical Society Monographs. New Series, The Clarendon Press Oxford University Press, New York (1991), Oxford Science Publications. MR 1150978 (93h:32021)
  • 15. I. Laine, Nevanlinna theory and complex differential equations, Vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin (1993). MR 1207139 (94d:34008)
  • 16. M. J. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3, 3 (1983), 351-385. MR 0741393 (85k:58049)
  • 17. R. Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical systems, Valparaiso 1986, Vol. 1331 of Lecture Notes in Math., Springer, Berlin (1988), 86-117. MR 0961095 (90j:58073)
  • 18. S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic dynamics, Vol. 66 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2000), Translated from the 1995 Japanese original and revised by the authors. MR 1747010 (2002c:37064)
  • 19. J. Noguchi and T. Ochiai, Geometric function theory in several complex variables, Vol. 80 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI (1990), Translated from the Japanese by Noguchi. MR 1084378 (92e:32001)
  • 20. Y. Okuyama, Nevanlinna, Siegel, and Cremer, Indiana Univ. Math. J. 53, 3 (2004), 755-763. MR 2086699
  • 21. Y. Okuyama, Complex dynamics, value distributions, and potential theory, Ann. Acad. Sci. Fenn. Math. 30, 2 (2005), 303-311. MR 2173366
  • 22. Y. Okuyama, Valiron, Nevanlinna and Picard exceptional sets of iterations of rational functions, Proc. Japan Acad. Ser. A Math. Sci. 81 2 (2005), 23-26. MR 2126072 (2006a:30026)
  • 23. B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Comm. Math. Phys. 200, 3 (1999), 661-683. MR 1675133 (2001j:32018)
  • 24. M. Sodin, Value distribution of sequences of rational functions, Entire and subharmonic functions, Vol. 11 of Adv. Soviet Math., Amer. Math. Soc., Providence, RI (1992), 7-20. MR 1188001 (93k:30050)
  • 25. H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity 13, 4 (2000), 995-1019. MR 1767945 (2001g:37060)
  • 26. H. Sumi, unpublished manuscript (2001).
  • 27. P. Tortrat, Aspects potentialistes de l'itération des polynômes, Séminaire de Théorie du Potentiel, Paris, No. 8, Vol. 1235 of Lecture Notes in Math., Springer, Berlin (1987), 195-209.
  • 28. T. Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46, 3 (1994), 545-555. MR 1276837 (95d:32030)

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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.

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