Brolin's theorem for curves in two complex dimensions

[Théorème de Brolin pour les courbes en dimension deux]

Favre, Charles ; Jonsson, Mattias

Annales de l'Institut Fourier, Tome 53 (2003), p. 1461-1501 / Harvested from Numdam

Résumé

Pour toute application $f:{\mathbb {P}}^2\rightarrow {\mathbb {P}}^2$ de degré $d\ge 2$ nous donnons des conditions suffisantes sur un courant positif fermé $S$ de bidegré $(1,1)$, pour que la suite $d^{-n} f^{n*} S$ converge vers le courant de Green lorsque $n\rightarrow \infty $. Nous conjecturons aussi des conditions nécessaires pour ce problème de convergence.

Abstract

Given a holomorphic mapping $f:{\mathbb {P}}^2\rightarrow {\mathbb {P}}^2$ of degree $d\ge 2$ we give sufficient conditions on a positive closed (1,1) current of $S$ of unit mass under which $d^{-n}f^{n*}S$ converges to the Green current as $n\rightarrow \infty $. We also conjecture necessary condition for the same convergence.

Détail

Publié le : 2003-01-01

MR 2032940 | Zbl 02014683 | 4 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1985
Classification:  37F10,  32U25
Mots clés: dynamique holomorphe, courants, nombre de Lelong, équidistribution, nombre de Kiselman, estimations de volume, multiplicités asymptotiques

Bibliographie

[B] H. Brolin Invariant sets under iteration of rational functions, Ark. Mat, Tome 6 (1965), pp. 103-144 | Article | MR 194595 | Zbl 0127.03401

[BD] J.-Y. Briend; J. Duval Deux caractérisations de la mesure d'équilibre d'un endomorphisme de $P^k(C)$ (to appear in Publ. Math. IHES.) | Numdam | Zbl 1010.37004

[BS] E. Bedford; J. Smillie Fatou-Bieberbach domains arising from polynomial diffeomorphisms, Indiana Math. J, Tome 40 (1991), pp. 789-792 | Article | MR 1119197 | Zbl 0739.32027

[CL] D. Cerveau; A. Lins; Neto . ph{{Hypersurfaces exceptionnelles des endomorphismes de $P^n$, Bol. Soc. Brasil. Mat. (N.S.), Tome 31 (2000), pp. 155-161 | Article | MR 1785266 | Zbl 0967.32022

[D] M. Dabija Algebraic and geometric dynamics in several complex variables (2000) (PhD thesis, University of Michigan)

[D1] J.-P. Demailly; V. Ancona And A. Silva, Editors Monge-Ampère operators, Lelong numbers and intersection theory., Complex analysis and geometry, Plenum Press (Univ. Ser. Math) (1993), pp. 115-193 | Zbl 0792.32006

[D2] J.-P. Demailly Pseudoconvex-concave duality and regularization of currents., Several complex variables (Berkeley, CA, 1995-1996), Cambridge Univ. Press, Cambridge (1999), pp. 233-271 | Zbl 0960.32011

[DJ] M. Dabija; M. Jonsson Holomorphic mappings of $P^2$ preserving a family of curves (preprint.)

[F1] C. Favre Note on pull-back and Lelong number of currents, Bull. Soc. Math. France, Tome 127 (1999), pp. 445-458 | Numdam | MR 1724404 | Zbl 0937.32005

[F2] C. Favre Dynamique des applications rationnelles (2000) (PhD thesis, Université de Paris Sud, Orsay)

[F3] C. Favre Multiplicity of holomorphic functions, Math. Ann, Tome 316 (2000), pp. 355-378 | Article | MR 1741274 | Zbl 0948.32020

[FG] C. Favre; V. Guedj Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Math. J., Tome 50 (2001), pp. 881-934 | MR 1871393 | Zbl 1046.37026

[FLM] A. Freire; A. Lopez; R. Mañé An invariant measure for rational maps, Bol. Soc. Bras. Mat., Tome 14 (1983), pp. 45-62 | Article | MR 736568 | Zbl 0568.58027

[FS1] J. E. Fornæss; N. Sibony Complex Hénon mappings in $C^2$ and Fatou-Bieberbach domains, Duke Math. J., Tome 65 (1992), pp. 345-380 | Article | MR 1150591 | Zbl 0761.32015

[FS2] J. E. Forn\Aess; N. Sibony; P. M. Gauthier And G. Sabidussi, Editors Complex dynamics in higher dimension., Complex Potential Theory, Kluwer, Dordrecht (1994), pp. 131-186

[FS3] J. E. Forn\Aess; N. Sibony Complex dynamics in higher dimension I, Astérisque, Tome 222 (1994), pp. 201-231 | MR 1285389 | Zbl 0813.58030

[FS4] J. E. Forn\Aess; N. Sibony; T. Bloom Et Al., Editors Complex dynamics in higher dimension II., Modern Methods in Complex Analysis, Princeton University Press (Annals of Mathematics Studies) Tome 137 (1995), pp. 135-182 | Zbl 0847.58059

[H] L. Hörmander Introduction to complex analysis in several variables, North Holland (1990) | MR 1045639 | Zbl 0685.32001

[K1] C. O. Kiselman Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math, Tome 60 (1994), pp. 173-197 | MR 1301603 | Zbl 0827.32016

[K2] C. O. Kiselman Ensembles de sous-niveau et images inverses des fonctions plurisousharmoniques, Bull. Sci. Math., Tome 124 (2000), pp. 75-92 | Article | MR 1742495 | Zbl 0955.32025

[Kl] M. Klimek Pluripotential theory, Oxford Science Publications, London Mathematical Society Monographs, New Series (1991) | MR 1150978 | Zbl 0742.31001

[L] M. Lyubich Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, Tome 3 (1983), pp. 351-385 | MR 741393 | Zbl 0537.58035

[Lo] S. \Lojasiewicz Introduction to complex analytic geometry, Birkhäuser Verlag, Basel (1991) | MR 1131081 | Zbl 0747.32001

[M] S.K. Mimouni Singularités des fonctions plurisousharmoniques et courants de Liouville (2001) (Thèse de la faculté des sciences de Monastir, Tunisie)

[RS] A. Russakovskii; B. Shiffman Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J, Tome 46 (1997), pp. 897-932 | Article | MR 1488341 | Zbl 0901.58023

[S] H. Skoda Sous-ensembles analytiques d'ordre fini ou infini dans $\mathbb{C}^n$, Bull. Soc. Math. France, Tome 100 (1972), pp. 353-408 | Numdam | MR 352517 | Zbl 0246.32009

[SSU] B. Shiffman; M. Shishikura; T. Ueda Totally invariant curves on $\mathbb{P}^2$ (preprint)