Geometry of currents, intersection theory and dynamics of horizontal-like maps
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 423-457.

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.

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.

DOI: 10.5802/aif.2188
Classification: 37F,  32H50,  32U40
Keywords: Structural discs of currents, Green current, equilibrium measure, mixing, entropy.
Dinh, Tien-Cuong 1; Sibony, Nessim 2

1 Institut de Mathématique de Jussieu Plateau 7D, Analyse Complexe 175 rue du Chevaleret 75013 Paris (France)
2 Université Paris-Sud Mathématique - Bâtiment 425 UMR 8628 91405 Orsay (France)
@article{AIF_2006__56_2_423_0,
     author = {Dinh, Tien-Cuong and Sibony, Nessim},
     title = {Geometry of currents, intersection theory and dynamics of horizontal-like maps},
     journal = {Annales de l'Institut Fourier},
     pages = {423--457},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     doi = {10.5802/aif.2188},
     mrnumber = {2226022},
     zbl = {1089.37036},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2188/}
}
TY  - JOUR
AU  - Dinh, Tien-Cuong
AU  - Sibony, Nessim
TI  - Geometry of currents, intersection theory and dynamics of horizontal-like maps
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 2006///
SP  - 423
EP  - 457
VL  - 56
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2188/
UR  - https://www.ams.org/mathscinet-getitem?mr=2226022
UR  - https://zbmath.org/?q=an%3A1089.37036
UR  - https://doi.org/10.5802/aif.2188
DO  - 10.5802/aif.2188
LA  - en
ID  - AIF_2006__56_2_423_0
ER  - 
%0 Journal Article
%A Dinh, Tien-Cuong
%A Sibony, Nessim
%T Geometry of currents, intersection theory and dynamics of horizontal-like maps
%J Annales de l'Institut Fourier
%D 2006
%P 423-457
%V 56
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2188
%R 10.5802/aif.2188
%G en
%F AIF_2006__56_2_423_0
Dinh, Tien-Cuong; Sibony, Nessim. Geometry of currents, intersection theory and dynamics of horizontal-like maps. Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 423-457. doi : 10.5802/aif.2188. http://archive.numdam.org/articles/10.5802/aif.2188/

[1] Bedford, E.; Lyubich, M.; Smillie, J. Polynomial diffeomorphisms of 2 , V: The measure of maximal entropy and laminar currents, Invent. Math., Volume 112 (1993) no. 1, pp. 77-125 | DOI | Zbl

[2] Bedford, E.; Smillie, J. Polynomial diffeomorphisms of 2 , III: Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann., Volume 294 (1992), pp. 395-420 | DOI | Zbl

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

[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 2 , Math. Ann., Volume 333 (2005) no. 4, pp. 703-739 | DOI | Zbl

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

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

[8] Dinh, T. C.; Sibony, N. Dynamics of regular birational maps in k , J. Funct. Anal., Volume 222 (2005) no. 1, pp. 202-216 | DOI | Zbl

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

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

[11] Dujardin, R. Hénon-like mappings in 2 , Amer. J. Math., Volume 126 (2004), pp. 439-472 | DOI | Zbl

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

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

[14] Fornæss, J. E.; Sibony, N. Complex Hénon mappings in 2 and Fatou-Bieberbach domains, Duke Math. J., Volume 65 (1992), pp. 345-380 | DOI | Zbl

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

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

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

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

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

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

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

[22] Sibony, N. Dynamique des applications rationnelles de k , Panoramas et Synthèses, Volume 8 (1999), pp. 97-185 | Zbl

[23] Smillie, J. The entropy of polynomial diffeomorphisms of 2 , Ergodic Theory & Dynamical Systems, Volume 10 (1990), pp. 823-827 | Zbl

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

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

Cited by Sources: