Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T09:25:53.078Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-Archimedean Montel’s theorem

Part of: Non-Archimedean analysis Arithmetic and non-Archimedean dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  21 February 2012

Charles Favre ,
Jan Kiwi  and
Eugenio Trucco
Charles Favre
Affiliation:
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, Cedex, France (email: [email protected])
Jan Kiwi
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile (email: [email protected])
Eugenio Trucco
Affiliation:
Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral de Chile, Casilla 567, Valdivia, Chile (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a version of Montel’s theorem for analytic functions over a non-Archimedean complete valued field. We propose a definition of normal family in this context, and give applications of our results to the dynamics of non-Archimedean entire functions.

Keywords

normal familyBerkovich spacesnon-Archimedean analysis

MSC classification

Secondary: 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 37P50: Dynamical systems on Berkovich spaces 30D45: Bloch functions, normal functions, normal families
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 966 - 990
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AB64]Atiyah, M. and Bott, R., On the Woods Hole fixed point theorem, in Proceedings of the Woods Hole conference on algebraic geometry (1964), Available at: www.jmilne.org/math/Documents/woodshole3.pdf.Google Scholar
[Bak63]Baker, I. N., Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206–214.CrossRefGoogle Scholar
[BR10]Baker, M. and Rumely, R., Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, vol. 159 (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
[Ben98]Benedetto, V., Fatou components in p-adic dynamics, Thesis, Brown University (1998).Google Scholar
[Ber90]Berkovich, V., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
[B{é}z01]Bézivin, J.-P., Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques, Ann. Inst. Fourier (Grenoble) 51 (2001), 1635–1661.CrossRefGoogle Scholar
[Ere89]Eremenko, A. E., On the iteration of entire functions, Dynamical Systems and Ergodic Theory, Banach Center Publications, vol. 23 (Polish Scientific, Warsaw, 1989), 339–345.Google Scholar
[Fab11]Faber, X., Topology and geometry of the Berkovich ramification locus for rational functions, Preprint (2011), arXiv:1102.1432.Google Scholar
[Fav11]Favre, C., Countability properties of some Berkovich spaces, Preprint (2011), arXiv:1103.6233.Google Scholar
[FJ04]Favre, C. and Jonsson, M., The valuative tree, Lecture Notes in Mathematics, vol. 1853 (Springer, Berlin, 2004).CrossRefGoogle Scholar
[FR10]Favre, C. and Rivera-Letelier, J., Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3) 100 (2010), 116–154.CrossRefGoogle Scholar
[FV04]Fresnel, J. and Van der Put, M., Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218 (Birkhäuser, Basel, 2004).CrossRefGoogle Scholar
[SGA5]Grothendieck, A., Formule de Lefshetz, in Exposé III du séminaire de géométrie algébrique du Bois-Marie 1965–1966 (SGA 5), Edité par Luc Illusie, Lecture Notes in Mathematics, vol. 589 (Springer, Berlin–New York, 1977).Google Scholar
[Hsi00]Hsia, L.-C., Closure of periodic points over a non-Archimedean field, J. Lond. Math. Soc. (2) 62 (2000), 685–700.CrossRefGoogle Scholar
[HY00]Hu, P.-C. and Yang, C.-C., Meromorphic functions over non-Archimedean fields, Mathematics and its Applications, vol. 522 (Kluwer Academic, 2000).CrossRefGoogle Scholar
[KS08]Kisaka, M. and Shishikura, M., On multiply connected wandering domains of entire functions, in Transcendental dynamics and complex analysis, London Mathematical Society Lecture Note Series, vol. 348 (Cambridge University Press, Cambridge, 2008), 217–250.CrossRefGoogle Scholar
[Kra10]Krantz, S. G., On limits of sequences of holomorphic functions, Preprint (2010), arXiv:1010.1285.Google Scholar
[Mar31]Marty, F., Recherches sur la répartition des valeurs d’une fonction méromorphe, Ann. Fac. Sci. Univ. Toulouse (3) 23 (1931), 183–261.CrossRefGoogle Scholar
[Poi11]Poineau, J., Les espaces de Berkovich sont angéliques, Preprint (2011), arXiv:1105.0250.Google Scholar
[Riv03]Rivera-Letelier, J., Dynamique des fonctions rationnelles sur des corps locaux, in Geometric methods in dynamics. II, Astérisque, vol. 287 (2003), xv, 147–230.Google Scholar
[Zal98]Zalcman, L., Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215–230.CrossRefGoogle Scholar