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Dynamics of regular birational maps in \(\mathbb P^k\). (English) Zbl 1067.37055
Summary: We study some spaces of currents of bidegree \((p,p)\). As an application, we construct the equilibrium measure for a large class of birational maps of \(\mathbb P^k\), as intersection of Green currents. We show that these currents are extremal and that the corresponding measure is mixing.

MSC:
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37A25 Ergodicity, mixing, rates of mixing
32U40 Currents
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[1] Bedford, E.; Smillie, J., Polynomial diffeomorphisms of \(\mathbb{C}^2\) III, Math. ann., 294, 395-420, (1992) · Zbl 0765.58013
[2] Bedford, E.; Lyubich, M.; Smillie, J., Polynomial diffeomorphisms of \(\mathbb{C}^2\) IV, Invent. math., 112, 1, 77-125, (1993) · Zbl 0792.58034
[3] Bost, J.B.; Gillet, H.; Soulé, C., Heights of projective varieties and positive Green forms, J. amer. math. soc., 7, 4, 903-1027, (1994) · Zbl 0973.14013
[4] Demailly, J.P., Regularization of closed positive currents and intersection theory, J. algebraic geom., 1, 361-409, (1992) · Zbl 0777.32016
[5] T.C. Dinh, Suites d’applications méromorphes multivaluées et courants laminaires, preprint, 2003, arxiv.org/abs/math.DS/0309421.
[6] T.C. Dinh, R. Dujardin, N. Sibony, On the dynamics near infinity of some polynomial mappings in \(\mathbb{C}^2\), preprint, 2004, arxiv.org/abs/math.DS/0407451. · Zbl 1079.37040
[7] Dinh, T.C.; Sibony, N., Dynamique des applications d’allure polynomiale, J. math. pures appl., 82, 367-423, (2003) · Zbl 1033.37023
[8] Dinh, T.C.; Sibony, N., Dynamique des applications semi-régulières, Arkiv för mat., 42, 61-85, (2004) · Zbl 1059.37033
[9] T.C. Dinh, N. Sibony, Distribution des valeurs de transformations méromorphes et applications, preprint, 2003, arxiv.org/abs/math.DS/0306095.
[10] T.C. Dinh, N. Sibony, Green currents for automorphisms of compact Kähler manifolds, J. Amer. Math. Sco., to appear. · Zbl 1066.32024
[11] T.C. Dinh, N. Sibony, Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math., to appear. · Zbl 1084.54013
[12] Fornæss, J.E.; Sibony, N., Complex Hénon mappings in \(\mathbb{C}^2\) and fatou – bieberbach domains, Duke math. J., 65, 345-380, (1992) · Zbl 0761.32015
[13] Fornæss, J.E.; Sibony, N., Complex dynamics in higher dimension, (), 131-186 · Zbl 0811.32019
[14] Fornæss, J.E.; Sibony, N., Oka’s inequality for currents and applications, Math. ann., 301, 399-419, (1995) · Zbl 0832.32010
[15] V. Guedj, Courants extrémaux et dynamique complexe, prépublication, 2004.
[16] Guedj, V.; Sibony, N., Dynamics of polynomial automorphisms of \(\mathbb{C}^k\), Arkiv för mat., 40, 207-243, (2002) · Zbl 1034.37025
[17] Sibony, N., Dynamique des applications rationnelles de \(\mathbb{P}^k\), Panoram. synthèses, 8, 97-185, (1999) · Zbl 1020.37026
[18] Skoda, H., Prolongement des courants positifs, fermés de masse finie, Invent. math., 66, 361-376, (1982) · Zbl 0488.58002
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