Non-Archimedean normal families
Annales de l'Institut Fourier, Volume 71 (2021) no. 4, pp. 1677-1732.

We present several results on the compactness of the space of morphisms between analytic spaces in the sense of Berkovich. We show that under certain conditions on the source, every sequence of analytic maps having an affinoid target has a subsequence that converges pointwise to a continuous map. We also study the class of continuous maps that arise in this way. Locally, they turn to be analytic after a certain base change. Our results naturally lead to a definition of normal families. We give some applications to the dynamics of an endomorphism of the projective space. We introduce two natural notions of Fatou set and generalize to the non-Archimedan setting a theorem of Ueda stating that every Fatou component is hyperbolically imbedded in the projective space.

Nous présentons plusieurs résultats concernant la compacité de l’espace des morphismes entre espaces analytiques au sens de Berkovich. Nous montrons que sous certaines conditions sur l’espace source, toute suite d’applications analytiques à valeurs dans un espace affinoïde admet une sous-suite qui converge ponctuellement vers une application continue. Nous étudions aussi la classe des applications continues qui apparaissent comme de telles limites. Localement ces applications deviennent analytiques après changement de base. Nos résultats amènent naturellement à la notion de familles normales. Nous donnons quelques applications à la dynamique des endomorphismes de l’espace projectif. Nous introduisons deux notions naturelles d’ensemble de Fatou et généralisons dans le cadre non-Archimédien un théorème de Ueda qui stipule que toute composante de Fatou est hyperboliquement plongée dans l’espace projectif.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/aif.3432
Classification: 32P05, 37P50, 32A19
Keywords: normal families, Berkovich spaces
Mot clés : famille normale, espace de Berkovich
Rodríguez Vázquez, Rita 1

1 CMLS École polytechnique Institut Polytechnique de Paris 91128 Palaiseau cedex (France)
@article{AIF_2021__71_4_1677_0,
     author = {Rodr{\'\i}guez V\'azquez, Rita},
     title = {Non-Archimedean normal families},
     journal = {Annales de l'Institut Fourier},
     pages = {1677--1732},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {4},
     year = {2021},
     doi = {10.5802/aif.3432},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3432/}
}
TY  - JOUR
AU  - Rodríguez Vázquez, Rita
TI  - Non-Archimedean normal families
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 1677
EP  - 1732
VL  - 71
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3432/
DO  - 10.5802/aif.3432
LA  - en
ID  - AIF_2021__71_4_1677_0
ER  - 
%0 Journal Article
%A Rodríguez Vázquez, Rita
%T Non-Archimedean normal families
%J Annales de l'Institut Fourier
%D 2021
%P 1677-1732
%V 71
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3432/
%R 10.5802/aif.3432
%G en
%F AIF_2021__71_4_1677_0
Rodríguez Vázquez, Rita. Non-Archimedean normal families. Annales de l'Institut Fourier, Volume 71 (2021) no. 4, pp. 1677-1732. doi : 10.5802/aif.3432. http://www.numdam.org/articles/10.5802/aif.3432/

[1] Baker, Matthew; Rumely, Robert Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, 159, American Mathematical Society, 2010, xxxiv+428 pages | DOI | MR

[2] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR

[3] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci. (1994) no. 78, pp. 5-161 | MR | Zbl

[4] Bosch, Siegfried Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume, Math. Ann., Volume 229 (1977) no. 1, pp. 25-45 | DOI | MR | Zbl

[5] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, 261, Springer, 1984 | Zbl

[6] Brody, Robert Compact manifolds and hyperbolicity, Trans. Am. Math. Soc., Volume 235 (1978), pp. 213-219 | MR | Zbl

[7] Chambert-Loir, Antoine Heights and measures on analytic spaces. A survey of recent results, and some remarks, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II (London Mathematical Society Lecture Note Series), Volume 384, Cambridge University Press, 2011, pp. 1-50 | MR | Zbl

[8] Cherry, William; Ru, Min Rigid analytic Picard theorems, Am. J. Math., Volume 126 (2004) no. 4, pp. 873-889 | DOI | MR | Zbl

[9] Ducros, Antoine Espaces de Berkovich, polytopes, squelettes et théorie des modèles, Confluentes Math., Volume 4 (2012) no. 4, 1250007, 57 pages | MR | Zbl

[10] Escassut, Alain; Mai Netti, Nicolas Shilov boundary for normed algebras, Topics in analysis and its applications (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 147, Kluwer Academic Publishers, 2004, pp. 1-10 | DOI | MR | Zbl

[11] Favre, Charles; Kiwi, Jan; Trucco, Eugenio A non-Archimedean Montel’s theorem, Compos. Math., Volume 148 (2012) no. 3, pp. 966-990 | DOI | MR | Zbl

[12] Favre, Charles; Rivera-Letelier, Juan Théorème d’équidistribution de Brolin en dynamique p-adique, C. R. Math. Acad. Sci. Paris, Volume 339 (2004) no. 4, pp. 271-276 | DOI | MR | Zbl

[13] Favre, Charles; Rivera-Letelier, Juan Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., Volume 335 (2006) no. 2, pp. 311-361 | DOI | MR | Zbl

[14] Favre, Charles; Rivera-Letelier, Juan Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc., Volume 100 (2010) no. 1, pp. 116-154 | DOI | MR | Zbl

[15] Fornaess, John Erik; Sibony, Nessim Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies), Volume 137, Princeton University Press, 1995, pp. 135-182 | MR | Zbl

[16] Fresnel, Jean; van der Put, Marius Rigid analytic geometry and its applications, Progress in Mathematics, 218, Birkhäuser, 2004, xii+296 pages | DOI | MR

[17] Hsia, Liang-Chung Closure of periodic points over a non-Archimedean field, J. Lond. Math. Soc. (2000) no. 62, pp. 685-700 | DOI | MR | Zbl

[18] Hubbard, John H. The Hénon mapping in the complex domain, Chaotic dynamics and fractals (Atlanta, 1985) (Notes and Reports in Mathematics in Science and Engineering), Volume 2, Academic Press Inc., 1986, pp. 101-111 | DOI | MR | Zbl

[19] Jonsson, Mattias Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications (Lecture Notes in Mathematics), Volume 2119, Springer, 2015, pp. 205-366 | DOI | MR | Zbl

[20] Kawaguchi, Shu; Silverman, Joseph H. Dynamics of projective morphisms having identical canonical heights, Proc. Lond. Math. Soc., Volume 95 (2007) no. 2, pp. 519-544 | DOI | MR | Zbl

[21] Kawaguchi, Shu; Silverman, Joseph H. Non-Archimedean Green functions and dynamics on projective space, Math. Z., Volume 262 (2009) no. 1, pp. 173-197 | DOI | MR | Zbl

[22] Kiehl, Reinhardt Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Invent. Math., Volume 2 (1967), pp. 256-273 | DOI | MR | Zbl

[23] Kobayashi, Shoshichi Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer, 1998 | DOI

[24] Lang, Serge Introduction to Complex Hyperbolic Spaces, Springer, 1987 | DOI

[25] Maegawa, Kazutoshi Fatou sets for rational maps of k , Mich. Math. J., Volume 52 (2004) no. 1, pp. 3-11 | DOI | MR | Zbl

[26] Montel, Paul Sur les suites infinies de fonctions, Ann. Sci. Éc. Norm. Supér., Volume 24 (1907), pp. 233-334 | DOI | Numdam | MR | Zbl

[27] Poineau, Jérôme Les espaces de Berkovich sont angéliques, Bull. Soc. Math. Fr., Volume 141 (2013) no. 2, pp. 267-297 | DOI | Numdam | Zbl

[28] Poineau, Jérôme Sur les composantes connexes d’une famille d’espaces analytiques p-adiques, Forum Math. Sigma, Volume 2 (2014), e14, 21 pages | DOI | MR | Zbl

[29] van der Put, Marius The class group of a one-dimensional affinoid space, Ann. Inst. Fourier, Volume 30 (1980) no. 4, pp. 155-164 | Numdam | MR | Zbl

[30] Sibony, Nessim Dynamique des applications rationnelles de P k , Dynamique et géométrie complexes (Lyon, 1997) (Panoramas et Synthèses), Volume 8, Société Mathématique de France, 1999, p. ix-x, xi–xii, 97–185 | MR | Zbl

[31] Silverman, Joseph H. The arithmetic of dynamical systems, Graduate Texts in Mathematics, 241, Springer, 2007, x+511 pages | DOI | MR

[32] Temkin, Michael On local properties of non-Archimedean analytic spaces, Math. Ann., Volume 318 (2000) no. 3, pp. 585-607 | DOI | MR | Zbl

[33] Temkin, Michael On local properties of non-Archimedean analytic spaces. II, Isr. J. Math., Volume 140 (2004), pp. 1-27 | DOI | MR | Zbl

[34] Temkin, Michael Introduction to Berkovich analytic spaces, Bekovich spaces and applications (Ducros, Antoine; Favre, Charles; Nicaise, Johannes, eds.) (Lecture Notes in Mathematics), Volume 2119, Springer (2015), pp. 3-66 | MR

[35] Thuillier, Amaury Potential theory on curves in non-Archimedean geometry. Applications to Arakelov theory., Ph. D. Thesis, Université Rennes 1 (2005) (https://tel.archives-ouvertes.fr/tel-00010990)

[36] Ueda, Tetsuo Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan, Volume 46 (1994) no. 3, pp. 545-555 | DOI | MR | Zbl

[37] Zalcman, Lawrence A heuristic principle in complex function theory, Am. Math. Mon., Volume 82 (1975) no. 8, pp. 813-817 | DOI | MR | Zbl

Cited by Sources: