Equidistribution of preimages over nonarchimedean fields for maps of good reduction

Gignac, William

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1737-1779.

Dans cet article, nous montrons un analogue du théorème d’équidistribution des préimages en dynamique complexe pour des applications définies sur un corps non archimédien et ayant bonne réduction. Bien que, en général ce théorème ne soit qu’un analogue partiel, nous montrons que pour la plupart des applications ayant bonne réduction c’est un analogue exact. Dans le cas particulier où le corps non archimédien est muni de la norme triviale, nous montrons un résultat plus fort, à savoir que les préimages des valuations «  modérées  » sont équidistribuées asymptotiquement par rapport à une mesure canonique.

In this article we prove an analogue of the equidistribution of preimages theorem from complex dynamics for maps of good reduction over nonarchimedean fields. While in general our result is only a partial analogue of the complex equidistribution theorem, for most maps of good reduction it is a complete analogue. In the particular case when the nonarchimedean field in question is equipped with the trivial absolute value, we are able to supply a strengthening of the theorem, namely that the preimages of any tame valuation equidistribute to a canonical measure.

[1] Baker, Matthew H.; Rumely, Robert Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 3, pp. 625-688 | DOI | Numdam | MR | Zbl

[2] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990, pp. x+169 | MR | Zbl

[3] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. (1993) no. 78, p. 5-161 (1994) | DOI | Numdam | MR | Zbl

[4] Bourbaki, Nicolas Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998, pp. xxiv+625 (Translated from the French, Reprint of the 1989 English translation) | MR | Zbl

[5] Briend, Jean-Yves; Duval, Julien Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${\mathrm{P}}^k\left(ℂ\right)$, Publ. Math. Inst. Hautes Études Sci. (2001) no. 93, pp. 145-159 | DOI | Numdam | MR | Zbl

[6] Brolin, Hans Invariant sets under iteration of rational functions, Ark. Mat., Volume 6 (1965), p. 103-144 (1965) | DOI | MR | Zbl

[7] Chambert-Loir, Antoine Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | DOI | MR | Zbl

[8] Dinh, Tien-Cuong Analytic multiplicative cocycles over holomorphic dynamical systems, Complex Var. Elliptic Equ., Volume 54 (2009) no. 3-4, pp. 243-251 | DOI | MR | Zbl

[9] Dinh, Tien-Cuong; Sibony, Nessim Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), Volume 82 (2003) no. 4, pp. 367-423 | DOI | MR | Zbl

[10] Dinh, Tien-Cuong; Sibony, Nessim Equidistribution towards the Green current for holomorphic maps, Ann. Sci. Éc. Norm. Supér. (4), Volume 41 (2008) no. 2, pp. 307-336 | Numdam | MR | Zbl

[11] Edixhoven, B.; Evertse, J.-H. Diophantine approximation and abelian varieties, Lecture Notes in Mathematics, 1566, Springer-Verlag, Berlin, 1993, pp. xiv+127 (Introductory lectures, Papers from the conference held in Soesterberg, April 12–16, 1992) | DOI | MR

[12] Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math., Volume 125 (2003) no. 2, pp. 409-440 http://muse.jhu.edu.proxy.lib.umich.edu/journals/american_journal_of_mathematics/v125/125.2ein.pdf | DOI | MR | Zbl

[13] Eisenbud, David Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, pp. xvi+785 | MR | Zbl

[14] Faber, Xander Equidistribution of dynamically small subvarieties over the function field of a curve, Acta Arith., Volume 137 (2009) no. 4, pp. 345-389 | DOI | MR | Zbl

[15] Fakhruddin, Najmuddin Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc., Volume 18 (2003) no. 2, pp. 109-122 | MR | Zbl

[16] Favre, Charles; Jonsson, Mattias Brolin’s theorem for curves in two complex dimensions, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 5, pp. 1461-1501 | DOI | Numdam | MR | Zbl

[17] Favre, Charles; Jonsson, Mattias The valuative tree, Lecture Notes in Mathematics, 1853, Springer-Verlag, Berlin, 2004, pp. xiv+234 | DOI | MR | Zbl

[18] 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

[19] Favre, Charles; Rivera-Letelier, Juan Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3), Volume 100 (2010) no. 1, pp. 116-154 | DOI | MR | Zbl

[20] Fornæss, John Erik; Sibony, Nessim Complex dynamics in higher dimension. I, Astérisque (1994) no. 222, pp. 5, 201-231 Complex analytic methods in dynamical systems (Rio de Janeiro, 1992) | MR | Zbl

[21] Fornæss, John Erik; Sibony, Nessim Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) (Ann. of Math. Stud.), Volume 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 135-182 | MR | Zbl

[22] Freire, Alexandre; Lopes, Artur; Mañé, Ricardo An invariant measure for rational maps, Bol. Soc. Brasil. Mat., Volume 14 (1983) no. 1, pp. 45-62 | DOI | MR | Zbl

[23] Gignac, William Measures and dynamics on Noetherian spaces, 2012 (To appear in J. Geom. Anal. doi:10.1007/s12220-013-9394-9.) | MR

[24] Gubler, Walter Equidistribution over function fields, Manuscripta Math., Volume 127 (2008) no. 4, pp. 485-510 | DOI | MR | Zbl

[25] Guedj, Vincent Equidistribution towards the Green current, Bull. Soc. Math. France, Volume 131 (2003) no. 3, pp. 359-372 | Numdam | MR | Zbl

[26] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York, 1977, pp. xvi+496 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[27] Izumi, Shuzo A measure of integrity for local analytic algebras, Publ. Res. Inst. Math. Sci., Volume 21 (1985) no. 4, pp. 719-735 | DOI | MR | Zbl

[28] Jonsson, Mattias Dynamics on Berkovich spaces in low dimensions, 2012 (Prépub. arXiv:1201.1944)

[29] Jonsson, Mattias; Mustaţǎ, Mircea Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 6, pp. 2145-2209 (2013) MR3060755 | DOI | Numdam | MR | Zbl

[30] Lejeune-Jalabert, Monique; Teissier, Bernard Normal cones and sheaves of relative jets, Compositio Math., Volume 28 (1974), pp. 305-331 | Numdam | MR | Zbl

[31] Ljubich, Mikhail Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, Volume 3 (1983) no. 3, pp. 351-385 | DOI | MR | Zbl

[32] Majidi-Zolbanin, Mahdi; Miasnikov, Nikita; Szpiro, Lucien Entropy and flatness in local algebraic dynamics, Pub. Mat., Volume 57 (2013) no. 2, pp. 509-544 (doi:10.5565/PUBLMAT_57213_12) | DOI | MR

[33] Okuyama, Yûsuke Repelling periodic points and logarithmic equidistribution in non-archimedean dynamics, Acta Arith., Volume 152 (2012) no. 3, pp. 267-277 | DOI | MR

[34] Okuyama, Yûsuke Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics, Math. Z., Volume 273 (2013) no. 3-4, pp. 811-837 (MR3030679) | DOI | MR

[35] Parra, Rodrigo The Jacobian cocycle and equidistribution towards the Green current, 2011 (arXiv:1103.4633)

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

[37] Shafarevich, Igor R. Basic algebraic geometry. 1, Springer-Verlag, Berlin, 1994, pp. xx+303 (Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid) | MR | Zbl

[38] Sibony, Nessim Dynamique des applications rationnelles de ${\mathbb{P}}^k$, Dynamique et géométrie complexes (Lyon, 1997) (Panor. Synthèses), Volume 8, Soc. Math. France, Paris, 1999, pp. 97-185 | MR | Zbl

[39] Silverman, Joseph Moduli spaces and arithmetic dynamics, CRM monograph series, v. 30, American Mathematical Society, Providence, R.I., 2012 | MR | Zbl

[40] Szpiro, Lucien; Ullmo, Emmanuel; Zhang, Shou-Wu Équirépartition des petits points, Invent. Math., Volume 127 (1997) no. 2, pp. 337-347 | DOI | MR | Zbl

[41] Tougeron, Jean-Claude Idéaux de fonctions différentiables, Springer-Verlag, Berlin, 1972, pp. vii+219 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71) | MR | Zbl

[42] 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

[43] Yuan, Xinyi Big line bundles over arithmetic varieties, Invent. Math., Volume 173 (2008) no. 3, pp. 603-649 | DOI | MR | Zbl