The Journal of Geometric Analysis

, Volume 15, Issue 2, pp 207–227

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Article

Abstract

Let Fn: X1 → X2 be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by Fn and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points.

Let (Zn) be a sequence of holomorphic images of ℙs in a projective manifold. We prove that the currents, defined by integration on Zn, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ℂk or an automorphism of a projective manifold.

Math Subject Classifications

32U40 32H50 

Key Words and Phrases

Courant laminarité transformation méromorphe 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Références

  1. [1]
    Alexander, H. Projective capacity,Ann. Math. Studies 100, 3–27, (1981).Google Scholar
  2. [2]
    Bedford, E., Lyubich, M., and Smillie, J. Polynomial diffeomorphisms of ℂ2, IV. The measure of maximal entropy and laminar current,Invent. Math. 112, 77–125, (1993).CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    Briend, J. Y. and Duval, J. Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ℂℙk,Acta Math. 182, 143–157, (1999).CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Briend, J. Y. and Duval, J. Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ℙk(ℂ),IHES Publ. Math. 93, 145–159, (2001).CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    Cantat, S. Dynamique des automorphismes des surfaces K3,Acta Math. 187(1), 1–57, (2001).CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    de Thélin, H. Sur la laminanté de certains courants,Ann. Sci. Ecole Norm. Sup. (4) 37(2), 304–311, (2004).MathSciNetMATHGoogle Scholar
  7. [7]
    Diller, J. and Favre, C. Dynamics of bimeromorphic maps of surfaces,Amer. J. Math. 123, 1135–1169, (2001).CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Dinh, T. C. Distribution des préimages et des points périodiques d’une correspondance polynomiale, à paraître au,Bull. Soc. Math. France.Google Scholar
  9. [9]
    Dinh, T. C. and Sibony, N. Dynamique des applications d’allure polynomials,J. Math. Pures et Appl. 82, 367–423, (2003).MathSciNetMATHGoogle Scholar
  10. [10]
    Dinh, T. C. and Sibony, N. Distribution des valeurs de transformations méromorphes et applications, prépublication, (2003), arXiv:math.DS/0306095Google Scholar
  11. [11]
    Dinh, T. C. and Sibony, N. Green currents for holomorphic automorphisms of compact Kähler manifolds,J. Amer. Math. Soc. 18, 291–312, (2005).CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    Dujardin, R. Laminar currents in ℙ2,Math. Ann. 325, 745–765, (2003).CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Dujardin, R. Sur l’intersection des courants laminaires,Publicacions Mathematiques 48, 107–125, (2004).MathSciNetMATHGoogle Scholar
  14. [14]
    Dujardin, R. Laminar currents and entropy properties of surface birational maps, prépublication, (2003).Google Scholar
  15. [15]
    Duval, J. and Sibony, N. Polynomial convexity, rational convexity, and currents,Duke Math. J. 79, 487–513, (1995).CrossRefMathSciNetMATHGoogle Scholar
  16. [16]
    Federer, H.Geometric Measure Theory, Springer-Verlag Inc., New York, NY, (1969).MATHGoogle Scholar
  17. [17]
    Fornæss, J. E. and Sibony, N. Dynamics ofP 2 (examples). Laminations and foliations in dynamics, geometry and topology [Stony Brook, NY, 47–85, (1998)],Contemp. Math. 269, Amer. Math. Soc., Providence, RI, (2001).Google Scholar
  18. [18]
    Freire, A., Lopes, A.. and Mañé, R. An invariant measure for rational maps,Bol. Soc. Brasil. Mat. 14, 45–62, (1983).CrossRefMathSciNetMATHGoogle Scholar
  19. [19]
    Fulton, W.Intersection Theory, Springer-Verlag, (1984).Google Scholar
  20. [20]
    Griffiths, P. and Harris, J.Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, (1994).MATHGoogle Scholar
  21. [21]
    Gromov, M. Convex sets and Kahler manifolds,Advances in Differential Geometry and Topology, Word Sci. Publishing, Teaneck, NJ, 1–38, (1998).Google Scholar
  22. [22]
    Guedj, V. Ergodic properties of rational mappings with large topological degree,Ann. of Math., to appear.Google Scholar
  23. [23]
    Jonsson, M. and Weickert, B. A nonalgebraic attractor inP 2,Proc. Amer. Math. Soc. 128(10), 2999–3002, (2000).CrossRefMathSciNetMATHGoogle Scholar
  24. [24]
    Kirwan, F.Complex Algebraic Curves, London Mathematical Society Student Texts,23. Cambridge University Press, Cambridge, (1992).Google Scholar
  25. [25]
    Lyubich, M. J. Entropy properties of rational endomorphisms of the Riemann sphere,Ergodic Theory Dynam. Systems 3, 351–385, (1983).MathSciNetMATHGoogle Scholar
  26. [26]
    McMullen, C. T. Dynamics on K3 surfaces: Salem numbers and Siegel disks,J. Reine Angew. Math. 545, 201–233, (2002).MathSciNetMATHGoogle Scholar
  27. [27]
    Méo, M. Image inverse d’un courant positif fermé par une application surjective,C.R.A.S. 322, 1141–1144, (1996).MATHGoogle Scholar
  28. [28]
    Russakovskii, A. and Shiffman, B. Value distribution for sequences of rational mappings and complex dynamics,Indiana Univ. Math. J. 46, 897–932, (1997).CrossRefMathSciNetMATHGoogle Scholar
  29. [29]
    Sibony, N. Dynamique des applications rationnelles de ℙk,Panor. Synthèses 8, 97–185, (1999).MathSciNetGoogle Scholar
  30. [30]
    Sibony, N. and Wong, P. M. Some results on global analytic sets,Séminaire Lelong-Skoda, L.N.,822, 221–237, (1980).MathSciNetGoogle Scholar
  31. [31]
    Siu, Y. T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents,Invent. Math. 27, 53–156, (1974).CrossRefMathSciNetMATHGoogle Scholar
  32. [32]
    Skoda, H. Prolongement des courants positifs, fermés de masse finie,Invent. Math. 66, 361–376, (1982).CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2005

Authors and Affiliations

  1. 1.Mathématique - Bât. 425Université Paris-SudOrsayFrance

Personalised recommendations