Equidistribution towards the Green current for holomorphic maps
[Équidistribution selon le courant de Green pour les applications holomorphes]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 2, pp. 307-336.

Soient f un endomorphisme holomorphe non-inversible d’un espace projectif et f n son itéré d’ordre n. Nous prouvons que l’image réciproque par f n d’une hypersurface générique (au sens de Zariski), proprement normalisée, converge vers le courant de Green associé à f quand n tend vers l’infini. Nous donnons également un résultat analogue pour les images réciproques des (1,1)-courants positifs fermés et un résultat similaire pour les automorphismes polynomiaux réguliers de k .

Let f be a non-invertible holomorphic endomorphism of a projective space and f n its iterate of order n. We prove that the pull-back by f n of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to f when n tends to infinity. We also give an analogous result for the pull-back of positive closed (1,1)-currents and a similar result for regular polynomial automorphisms of  k .

DOI : 10.24033/asens.2069
Classification : 37F10, 32H50, 32U05
Keywords: Green current, exceptional set, plurisubharmonic function, Lelong number, regular automorphism
Mot clés : courants de Green, ensemble exceptionnel, fonction plurisousharmonique, nombre de Lelong, automorphisme régulier
@article{ASENS_2008_4_41_2_307_0,
     author = {Dinh, Tien-Cuong and Sibony, Nessim},
     title = {Equidistribution towards the {Green} current for holomorphic maps},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {307--336},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {2},
     year = {2008},
     doi = {10.24033/asens.2069},
     mrnumber = {2468484},
     zbl = {1160.32029},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2069/}
}
TY  - JOUR
AU  - Dinh, Tien-Cuong
AU  - Sibony, Nessim
TI  - Equidistribution towards the Green current for holomorphic maps
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2008
SP  - 307
EP  - 336
VL  - 41
IS  - 2
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2069/
DO  - 10.24033/asens.2069
LA  - en
ID  - ASENS_2008_4_41_2_307_0
ER  - 
%0 Journal Article
%A Dinh, Tien-Cuong
%A Sibony, Nessim
%T Equidistribution towards the Green current for holomorphic maps
%J Annales scientifiques de l'École Normale Supérieure
%D 2008
%P 307-336
%V 41
%N 2
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2069/
%R 10.24033/asens.2069
%G en
%F ASENS_2008_4_41_2_307_0
Dinh, Tien-Cuong; Sibony, Nessim. Equidistribution towards the Green current for holomorphic maps. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 2, pp. 307-336. doi : 10.24033/asens.2069. http://www.numdam.org/articles/10.24033/asens.2069/

[1] J.-Y. Briend & J. Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k (𝐂), Publ. Math. Inst. Hautes Études Sci. 93 (2001), 145-159. | Numdam | Zbl

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

[3] D. Cerveau & A. Lins Neto, Hypersurfaces exceptionnelles des endomorphismes de 𝐂P(n), Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), 155-161. | Zbl

[4] S. S. Chern, H. I. Levine & L. Nirenberg, Intrinsic norms on a complex manifold, in Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, 1969, 119-139. | Zbl

[5] J.-P. Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, in Complex analysis and geometry, Univ. Ser. Math., Plenum, 1993, 115-193. | MR | Zbl

[6] J.-P. Demailly, Complex analytic geometry, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, 2007.

[7] T.-C. Dinh, Distribution des préimages et des points périodiques d'une correspondance polynomiale, Bull. Soc. Math. France 133 (2005), 363-394. | Numdam | MR | Zbl

[8] T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires, J. Geom. Anal. 15 (2005), 207-227. | MR | Zbl

[9] T.-C. Dinh & N. Sibony, Dynamique des applications d'allure polynomiale, J. Math. Pures Appl. 82 (2003), 367-423. | Zbl

[10] T.-C. Dinh & N. Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (2006), 221-258. | Zbl

[11] T.-C. Dinh & N. Sibony, Pull-back of currents by holomorphic maps, Manuscripta Math. 123 (2007), 357-371. | Zbl

[12] T.-C. Dinh & N. Sibony, Super-potentials of positive closed currents, intersection theory and dynamics, preprint arXiv:math/0703702, 2007, to appear in Acta Math. | Zbl

[13] C. Favre & M. Jonsson, Brolin's theorem for curves in two complex dimensions, Ann. Inst. Fourier (Grenoble) 53 (2003), 1461-1501. | Zbl

[14] C. Favre & M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup. 40 (2007), 309-349. | Numdam | Zbl

[15] J. E. Fornæss & R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann. 248 (1980), 47-72. | Zbl

[16] J. E. Fornæss & N. Sibony, Complex Hénon mappings in 𝐂 2 and Fatou-Bieberbach domains, Duke Math. J. 65 (1992), 345-380. | Zbl

[17] J. E. Fornæss & N. Sibony, Complex dynamics in higher dimension. I. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992), Astérisque 222 (1994), 5, 201-231. | Zbl

[18] J. E. Fornæss & N. Sibony, Complex dynamics in higher dimension. II, in Modern methods in complex analysis (Princeton, NJ, 1992), Ann. of Math. Stud. 137, Princeton Univ. Press, 1995, 135-182. | Zbl

[19] A. Freire, A. Lopes & R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45-62. | Zbl

[20] V. Guedj, Equidistribution towards the Green current, Bull. Soc. Math. France 131 (2003), 359-372. | Numdam | MR | Zbl

[21] V. Guedj, Decay of volumes under iteration of meromorphic mappings, Ann. Inst. Fourier (Grenoble) 54 (2004), 2369-2386. | Numdam | MR | Zbl

[22] L. Hörmander, The analysis of linear partial differential operators. I, Grund. Math. Wiss. 256, Springer, 1983. | MR | Zbl

[23] P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Dunod, 1968. | Zbl

[24] M. J. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351-385. | MR | Zbl

[25] M. Meo, Image inverse d'un courant positif fermé par une application analytique surjective, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 1141-1144. | MR | Zbl

[26] M. Meo, Inégalités d'auto-intersection pour les courants positifs fermés définis dans les variétés projectives, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998), 161-184. | Numdam | MR | Zbl

[27] A. Russakovskii & B. Shiffman, Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J. 46 (1997), 897-932. | Zbl

[28] B. Shiffman, M. Shishikura & T. Ueda, On totally invariant varieties of holomorphic mappings of n , preprint, 2000.

[29] N. Sibony, Dynamique des applications rationnelles de 𝐏 k , in Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses 8, Soc. Math. France, 1999, 97-185. | MR | Zbl

[30] Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974), 53-156. | MR | Zbl

[31] J.-C. Tougeron, Idéaux de fonctions différentiables, 71, Springer, 1972. | Zbl

[32] T. Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46 (1994), 545-555. | MR | Zbl

[33] G. Vigny, Lelong-Skoda transform for compact Kähler manifolds and self-intersection inequalities, preprint arXiv:0711.3782v1, 2007. | MR | Zbl

Cité par Sources :