An Ahlfors Islands Theorem for non-archimedean meromorphic functions

Benedetto, Robert

doi:10.1090/S0002-9947-08-04546-7

An Ahlfors Islands Theorem for non-archimedean meromorphic functions
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Trans. Amer. Math. Soc. 360 (2008), 4099-4124 Request permission

Abstract:

We present a $p$-adic and non-archimedean version of Ahlfors’ Five Islands Theorem for meromorphic functions, extending an earlier theorem of the author for holomorphic functions. In the non-archimedean setting, the theorem requires only four islands, with explicit constants. We present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed.

References

W. W. Adams and E. G. Straus, Non-archimedian analytic functions taking the same values at the same points, Illinois J. Math. 15 (1971), 418–424. MR 277771
Lars Ahlfors, Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), no. 1, 157–194 (German). MR 1555403, DOI 10.1007/BF02420945
I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252–256. MR 226009, DOI 10.1007/BF01110294
Robert L. Benedetto, $p$-adic dynamics and Sullivan’s no wandering domains theorem, Compositio Math. 122 (2000), no. 3, 281–298. MR 1781331, DOI 10.1023/A:1002067315057
Robert L. Benedetto, Components and periodic points in non-Archimedean dynamics, Proc. London Math. Soc. (3) 84 (2002), no. 1, 231–256. MR 1863402, DOI 10.1112/plms/84.1.231
Robert L. Benedetto, Non-Archimedean holomorphic maps and the Ahlfors Islands theorem, Amer. J. Math. 125 (2003), no. 3, 581–622. MR 1981035
Walter Bergweiler, A new proof of the Ahlfors five islands theorem, J. Anal. Math. 76 (1998), 337–347. MR 1676971, DOI 10.1007/BF02786941
Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
Jean-Paul Bézivin, Sur les points périodiques des applications rationnelles en dynamique ultramétrique, Acta Arith. 100 (2001), no. 1, 63–74 (French). MR 1864626, DOI 10.4064/aa100-1-5
J.-P. Bézivin, Dynamique des fractions rationnelles $p$-adiques, monograph, 2005. Available online at http://www.math.unicaen.fr/~bezivin/dealatex.pdf
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
Abdelbaki Boutabaa, Théorie de Nevanlinna $p$-adique, Manuscripta Math. 67 (1990), no. 3, 251–269 (French, with English summary). MR 1046988, DOI 10.1007/BF02568432
Abdelbaki Boutabaa and Alain Escassut, Nevanlinna theory in characteristic $p$ and applications, Analysis and applications—ISAAC 2001 (Berlin), Int. Soc. Anal. Appl. Comput., vol. 10, Kluwer Acad. Publ., Dordrecht, 2003, pp. 97–107. MR 2022742, DOI 10.1007/978-1-4757-3741-7_{7}
William Cherry, A survey of Nevanlinna theory over non-Archimedean fields, Bull. Hong Kong Math. Soc. 1 (1997), no. 2, 235–249. International Workshop on Value Distribution Theory and Its Applications (Hong Kong, 1996). MR 1605198
William Cherry and Zhuan Ye, Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem, Trans. Amer. Math. Soc. 349 (1997), no. 12, 5043–5071. MR 1407485, DOI 10.1090/S0002-9947-97-01874-6
Capi Corrales-Rodrigáñez, Nevanlinna theory on the $p$-adic plane, Ann. Polon. Math. 57 (1992), no. 2, 135–147. MR 1182179, DOI 10.4064/ap-57-2-135-147
Jacques Dufresnoy, Sur les domaines couverts par les valeurs d’une fonction méromorphe ou algébroïde, Ann. Sci. École Norm. Sup. (3) 58 (1941), 179–259 (French). MR 0012669
Alain Escassut, Algèbres de Banach ultramétriques et algèbres de Krasner-Tate, Prolongement analytique et algèbres de Banach ultramétriques, Astérisque, No. 10, Sociéte Mathématique de France, Paris, 1973, pp. 1–107, 219 (French, with English summary). MR 0364676
A. Escassut, The ultrametric spectral theory, Period. Math. Hungar. 11 (1980), no. 1, 7–60. MR 571136, DOI 10.1007/BF02019983
Alain Escassut, Analytic elements in $p$-adic analysis, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370442, DOI 10.1142/9789812831019
David Goss, A short introduction to rigid analytic spaces, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 131–141. MR 1196516
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
Liang-Chung Hsia, Closure of periodic points over a non-Archimedean field, J. London Math. Soc. (2) 62 (2000), no. 3, 685–700. MR 1794277, DOI 10.1112/S0024610700001447
Pei-Chu Hu and Chung-Chun Yang, Meromorphic functions over non-Archimedean fields, Mathematics and its Applications, vol. 522, Kluwer Academic Publishers, Dordrecht, 2000. MR 1794326, DOI 10.1007/978-94-015-9415-8
Hà Huy Khoái and My Vinh Quang, On $p$-adic Nevanlinna theory, Complex analysis, Joensuu 1987, Lecture Notes in Math., vol. 1351, Springer, Berlin, 1988, pp. 146–158. MR 982080, DOI 10.1007/BFb0081250
Michel Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47–75 (French). MR 152519
Juan Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Astérisque 287 (2003), xv, 147–230 (French, with English and French summaries). Geometric methods in dynamics. II. MR 2040006
Juan Rivera-Letelier, Espace hyperbolique $p$-adique et dynamique des fonctions rationnelles, Compositio Math. 138 (2003), no. 2, 199–231 (French, with English summary). MR 2018827, DOI 10.1023/A:1026136530383
Philippe Robba, Fonctions analytiques sur les corps valués ultramétriques complets, Prolongement analytique et algèbres de Banach ultramétriques, Astérisque, No. 10, Société Mathématique de France, Paris, 1973, pp. 109–218, 219–220 (French, with English summary). MR 0357841
Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253, DOI 10.1007/978-1-4757-3254-2
Min Ru, A note on $p$-adic Nevanlinna theory, Proc. Amer. Math. Soc. 129 (2001), no. 5, 1263–1269. MR 1712881, DOI 10.1090/S0002-9939-00-05680-X
R. Rumely and M. Baker, Analysis and dynamics on the Berkovich projective line, preprint, 2004. Available online at arXiv:math.NT/0407433
Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 379852, DOI 10.2307/2319796