Equidistribution of preimages over nonarchimedean fields for maps of good reduction

William Gignac[1]

  • [1] Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, USA

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 4, page 1737-1779
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Gignac, William. "Equidistribution of preimages over nonarchimedean fields for maps of good reduction." Annales de l’institut Fourier 64.4 (2014): 1737-1779. <http://eudml.org/doc/275433>.

@article{Gignac2014,
abstract = {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.},
affiliation = {Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, USA},
author = {Gignac, William},
journal = {Annales de l’institut Fourier},
keywords = {equidistribution; nonarchimedean dynamics; Berkovich spaces; maps of good reduction; multiplicities; exceptional set; good reduction},
language = {eng},
number = {4},
pages = {1737-1779},
publisher = {Association des Annales de l’institut Fourier},
title = {Equidistribution of preimages over nonarchimedean fields for maps of good reduction},
url = {http://eudml.org/doc/275433},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Gignac, William
TI - Equidistribution of preimages over nonarchimedean fields for maps of good reduction
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 4
SP - 1737
EP - 1779
AB - 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.
LA - eng
KW - equidistribution; nonarchimedean dynamics; Berkovich spaces; maps of good reduction; multiplicities; exceptional set; good reduction
UR - http://eudml.org/doc/275433
ER -

References

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  1. Matthew H. Baker, Robert Rumely, Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble) 56 (2006), 625-688 Zbl1234.11082MR2244226
  2. Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 33 (1990), American Mathematical Society, Providence, RI Zbl0715.14013MR1070709
  3. Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. (1993), 5-161 (1994) Zbl0804.32019MR1259429
  4. Nicolas Bourbaki, Commutative algebra. Chapters 1–7, (1998), Springer-Verlag, Berlin Zbl0673.00001MR1727221
  5. Jean-Yves Briend, Julien Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k ( ℂ ) , Publ. Math. Inst. Hautes Études Sci. (2001), 145-159 Zbl1010.37004MR1863737
  6. Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144 (1965) Zbl0127.03401MR194595
  7. Antoine Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215-235 Zbl1112.14022MR2244803
  8. Tien-Cuong Dinh, Analytic multiplicative cocycles over holomorphic dynamical systems, Complex Var. Elliptic Equ. 54 (2009), 243-251 Zbl1161.37329MR2513537
  9. Tien-Cuong Dinh, Nessim Sibony, Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9) 82 (2003), 367-423 Zbl1033.37023MR1992375
  10. Tien-Cuong Dinh, Nessim Sibony, Equidistribution towards the Green current for holomorphic maps, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 307-336 Zbl1160.32029MR2468484
  11. Diophantine approximation and abelian varieties, 1566 (1993), EdixhovenB.B., Berlin MR1288998
  12. Lawrence Ein, Robert Lazarsfeld, Karen Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), 409-440 Zbl1033.14030MR1963690
  13. David Eisenbud, Commutative algebra, with a view toward algebraic geometry, 150 (1995), Springer-Verlag, New York Zbl0819.13001MR1322960
  14. Xander Faber, Equidistribution of dynamically small subvarieties over the function field of a curve, Acta Arith. 137 (2009), 345-389 Zbl1234.37058MR2506588
  15. Najmuddin Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), 109-122 Zbl1053.14025MR1995861
  16. Charles Favre, Mattias Jonsson, Brolin’s theorem for curves in two complex dimensions, Ann. Inst. Fourier (Grenoble) 53 (2003), 1461-1501 Zbl1113.32005MR2032940
  17. Charles Favre, Mattias Jonsson, The valuative tree, 1853 (2004), Springer-Verlag, Berlin Zbl1064.14024MR2097722
  18. Charles Favre, Juan Rivera-Letelier, Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), 311-361 Zbl1175.11029MR2221116
  19. Charles Favre, Juan Rivera-Letelier, Théorie ergodique des fractions rationnelles sur un corps ultramétrique, Proc. Lond. Math. Soc. (3) 100 (2010), 116-154 Zbl1254.37064MR2578470
  20. John Erik Fornæss, Nessim Sibony, Complex dynamics in higher dimension. I, Astérisque (1994), 5, 201-231 Zbl0813.58030MR1285389
  21. John Erik Fornæss, Nessim Sibony, Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) 137 (1995), 135-182, Princeton Univ. Press, Princeton, NJ Zbl0847.58059MR1369137
  22. Alexandre Freire, Artur Lopes, Ricardo Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45-62 Zbl0568.58027MR736568
  23. William Gignac, Measures and dynamics on Noetherian spaces, (2012) Zbl1320.37003MR3261717
  24. Walter Gubler, Equidistribution over function fields, Manuscripta Math. 127 (2008), 485-510 Zbl1189.14030MR2457191
  25. Vincent Guedj, Equidistribution towards the Green current, Bull. Soc. Math. France 131 (2003), 359-372 Zbl1070.37026MR2017143
  26. Robin Hartshorne, Algebraic geometry, (1977), Springer-Verlag, New York Zbl0531.14001MR463157
  27. Shuzo Izumi, A measure of integrity for local analytic algebras, Publ. Res. Inst. Math. Sci. 21 (1985), 719-735 Zbl0587.32016MR817161
  28. Mattias Jonsson, Dynamics on Berkovich spaces in low dimensions, (2012) Zbl06463429
  29. Mattias Jonsson, Mircea Mustaţǎ, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), 2145-2209 Zbl1272.14016MR3060755
  30. Monique Lejeune-Jalabert, Bernard Teissier, Normal cones and sheaves of relative jets, Compositio Math. 28 (1974), 305-331 Zbl0337.14013MR379897
  31. Mikhail Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351-385 Zbl0537.58035MR741393
  32. Mahdi Majidi-Zolbanin, Nikita Miasnikov, Lucien Szpiro, Entropy and flatness in local algebraic dynamics, Pub. Mat. 57 (2013), 509-544 Zbl1302.37059MR3114781
  33. Yûsuke Okuyama, Repelling periodic points and logarithmic equidistribution in non-archimedean dynamics, Acta Arith. 152 (2012), 267-277 Zbl1302.37069MR2885787
  34. Yûsuke Okuyama, Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics, Math. Z. 273 (2013), 811-837 Zbl1329.37085MR3030679
  35. Rodrigo Parra, The Jacobian cocycle and equidistribution towards the Green current, (2011) 
  36. Jérôme Poineau, Les espaces de Berkovich sont angéliques, Bull. Soc. Math. Fr. 141 (2013), 267-297 Zbl1314.14046MR3081557
  37. Igor R. Shafarevich, Basic algebraic geometry. 1, (1994), Springer-Verlag, Berlin Zbl0797.14001MR1328833
  38. Nessim Sibony, Dynamique des applications rationnelles de ℙ k , Dynamique et géométrie complexes (Lyon, 1997) 8 (1999), 97-185, Soc. Math. France, Paris Zbl1020.37026MR1760844
  39. Joseph Silverman, Moduli spaces and arithmetic dynamics, v. 30 (2012), American Mathematical Society, Providence, R.I. Zbl1247.37004MR2884382
  40. Lucien Szpiro, Emmanuel Ullmo, Shou-Wu Zhang, Équirépartition des petits points, Invent. Math. 127 (1997), 337-347 Zbl0991.11035MR1427622
  41. Jean-Claude Tougeron, Idéaux de fonctions différentiables, (1972), Springer-Verlag, Berlin Zbl0251.58001MR440598
  42. Tetsuo Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46 (1994), 545-555 Zbl0829.58025MR1276837
  43. Xinyi Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), 603-649 Zbl1146.14016MR2425137

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