Geometry of currents, intersection theory and dynamics of horizontal-like maps

[Géométrie des courants, théorie d’intersection et dynamique des applications d’allure horizontale]

Dinh, Tien-Cuong ; Sibony, Nessim

Annales de l'Institut Fourier, Tome 56 (2006), p. 423-457 / Harvested from Numdam

Résumé

Nous introduisons une géométrie sur le cône des courants positifs fermés de bidegré $(p,p)$ et nous l’utilisons pour définir l’intersection de tels courants. Nous construisons et étudions aussi les courants de Green et la mesure d’équilibre pour les applications d’allure horizontale, en toute dimension. Les courants de Green vérifient certaines propriétés d’extrémalité. La mesure d’équilibre est invariante, mélangeante et d’entropie maximale. Elle est égale à l’intersection des courants de Green associés à l’application et à son inverse.

Abstract

We introduce a geometry on the cone of positive closed currents of bidegree $(p,p)$ and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like mappings. The Green currents satisfy some extremality properties. The equilibrium measure is invariant, mixing and has maximal entropy. It is equal to the intersection of the Green currents associated to the horizontal-like map and to its inverse.

Détail

Publié le : 2006-01-01

MR 2226022 | Zbl 1089.37036 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2188
Classification:  37F,  32H50,  32U40
Mots clés: disque structurel de courants, courant de Green, mesure d’équilibre, mélange, entropie

Bibliographie

[1] Bedford, E.; Lyubich, M.; Smillie, J. Polynomial diffeomorphisms of $\mathbb{C}^2$, V: The measure of maximal entropy and laminar currents, Invent. Math., Tome 112 (1993) no. 1, pp. 77-125 | Article | Zbl 0792.58034

[2] Bedford, E.; Smillie, J. Polynomial diffeomorphisms of $\mathbb{C}^2$, III: Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann., Tome 294 (1992), pp. 395-420 | Article | Zbl 0765.58013

[3] Demailly, J. P. Monge-Ampère Operators, Lelong numbers and Intersection theory in Complex Analysis and Geometry, Plemum Press (1993), pp. 115-193 | Zbl 0792.32006

[4] Dinh, T. C. Decay of correlations for Hénon maps (to appear)

[5] Dinh, T. C.; Dujardin, R.; Sibony, N. On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$, Math. Ann., Tome 333 (2005) no. 4, pp. 703-739 | Article | Zbl 1079.37040

[6] Dinh, T. C.; Sibony, N. Dynamique des applications d’allure polynomiale, J. Math. Pures Appl., Tome 82 (2003), pp. 367-423 | Article | Zbl 1033.37023

[7] Dinh, T. C.; Sibony, N. Regularization of currents and entropy, Ann. Sci. Ecole Norm. Sup., Tome 37 (2004), pp. 959-971 | Numdam | Zbl 1074.53058

[8] Dinh, T. C.; Sibony, N. Dynamics of regular birational maps in $\mathbb{P}^k$, J. Funct. Anal., Tome 222 (2005) no. 1, pp. 202-216 | Article | Zbl 1067.37055

[9] Dinh, T. C.; Sibony, N. Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Amer. Math. Soc., Tome 18 (2005) no. 2, pp. 291-312 | Article | Zbl 1066.32024

[10] Dinh, T. C.; Sibony, N. Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math., Tome 161 (2005), pp. 1637-1644 | Article | Zbl 05004661

[11] Dujardin, R. Hénon-like mappings in $\mathbb{C}^2$, Amer. J. Math., Tome 126 (2004), pp. 439-472 | Article | Zbl 1064.37035

[12] Duval, J.; Sibony, N. Polynomial convexity, rational convexity, and currents, Duke Math. J., Tome 79 (1995) no. 2, pp. 487-513 | Article | Zbl 0838.32006

[13] Federer, H. Geometric Measure Theory, Springer Verlag, New York (1969) | Zbl 0176.00801

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

[15] Fornæss, J. E.; Sibony, N. Oka’s inequality for currents and applications, Math. Ann., Tome 301 (1995), pp. 399-419 | Article | Zbl 0832.32010

[16] Gromov, M. On the entropy of holomorphic maps, Enseignement Math., Tome 49 (2003), pp. 217-235 (Manuscript (1977)) | Zbl 1080.37051

[17] Harvey, R.; Polking, J. Extending analytic objects, Comm. Pure Appl. Math., Tome 28 (1975), pp. 701-727 | Article | Zbl 0323.32013

[18] Harvey, R.; Shiffman, B. A characterization of holomorphic chains, Ann. of Math. (2), Tome 99 (1974), pp. 553-587 | Article | Zbl 0287.32008

[19] Hörmander, L. The analysis of Linear partial differential operators I, Springer-Verlag (1983) | Zbl 0521.35001

[20] Katok, A.; Hasselblatt, B. Introduction to the modern theory of dynamical systems, Cambridge Univ. Press., Encycl. of Math. and its Appl., Tome 54 (1995) | Zbl 0878.58020

[21] Lelong, P. Fonctions plurisousharmoniques et formes différentielles positives, Dunod, Paris (1968) | Zbl 0195.11603

[22] Sibony, N. Dynamique des applications rationnelles de $\mathbb{P}^k$, Panoramas et Synthèses, Tome 8 (1999), pp. 97-185 | Zbl 1020.37026

[23] Smillie, J. The entropy of polynomial diffeomorphisms of $\mathbb{C}^2$, Ergodic Theory & Dynamical Systems, Tome 10 (1990), pp. 823-827 | Zbl 0695.58023

[24] Walters, P. An introduction to ergodic theory, Springer, Berlin-Heidelberg-New York (1982) | Zbl 0475.28009

[25] Yomdin, Y. Volume growth and entropy, Israel J. Math., Tome 57 (1987), pp. 285-300 | Article | Zbl 0641.54036