Equidistribution towards the Green current for holomorphic maps

Dinh, Tien-Cuong; Sibony, Nessim

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 ${ℂ}^k$.

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 ${ℂ}^k$.

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

[1] J.-Y. Briend & J. Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${\mathrm{P}}^k\left(𝐂\right)$, 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 $𝐂\mathrm{P}\left(n\right)$, 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 ${\mathbb{P}}^n$, 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