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Tien-Cuong Dinh; Nessim Sibony
Geometry of currents, intersection theory and dynamics of horizontal-like maps
(Géométrie des courants, théorie d’intersection et dynamique des applications d’allure horizontale)
Annales de l'institut Fourier, 56 no. 2 (2006), p. 423-457, doi: 10.5802/aif.2188
Article PDF | Reviews MR 2226022 | Zbl 1089.37036
Class. Math.: 37F, 32H50, 32U40
Keywords: Structural discs of currents, Green current, equilibrium measure, mixing, entropy.
[1] E. Bedford, M. Lyubich & J. Smillie, “Polynomial diffeomorphisms of $\mathbb{C}^2$, V: The measure of maximal entropy and laminar currents”, Invent. Math. 112 (1993) no. 1, p. 77-125
Article | Zbl 0792.58034
[2] E. Bedford & J. Smillie, “Polynomial diffeomorphisms of $\mathbb{C}^2$, III: Ergodicity, exponents and entropy of the equilibrium measure”, Math. Ann. 294 (1992), p. 395-420
Article | Zbl 0765.58013
[3] J. P. Demailly, “Monge-Ampère Operators, Lelong numbers and Intersection theory in Complex Analysis and Geometry”, Plemum Press (1993), p. 115-193 Zbl 0792.32006
[4] T. C. Dinh, “Decay of correlations for Hénon maps”, to appear
arXiv
[5] T. C. Dinh, R. Dujardin & N. Sibony, “On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$”, Math. Ann. 333 (2005) no. 4, p. 703-739
Article | Zbl 1079.37040
[6] T. C. Dinh & N. Sibony, “Dynamique des applications d’allure polynomiale”, J. Math. Pures Appl. 82 (2003), p. 367-423
Article | Zbl 1033.37023
[7] T. C. Dinh & N. Sibony, “Regularization of currents and entropy”, Ann. Sci. Ecole Norm. Sup. 37 (2004), p. 959-971
Numdam | Zbl 1074.53058
[8] T. C. Dinh & N. Sibony, “Dynamics of regular birational maps in $\mathbb{P}^k$”, J. Funct. Anal. 222 (2005) no. 1, p. 202-216
Article | Zbl 1067.37055
[9] T. C. Dinh & N. Sibony, “Green currents for holomorphic automorphisms of compact Kähler manifolds”, J. Amer. Math. Soc. 18 (2005) no. 2, p. 291-312
Article | Zbl 1066.32024
[10] T. C. Dinh & N. Sibony, “Une borne supérieure pour l’entropie topologique d’une application rationnelle”, Ann. of Math. 161 (2005), p. 1637-1644
Article | Zbl 05004661
[11] R. Dujardin, “Hénon-like mappings in $\mathbb{C}^2$”, Amer. J. Math. 126 (2004), p. 439-472
Article | Zbl 1064.37035
[12] J. Duval & N. Sibony, “Polynomial convexity, rational convexity, and currents”, Duke Math. J. 79 (1995) no. 2, p. 487-513
Article | Zbl 0838.32006
[13] H. Federer, Geometric Measure Theory, Springer Verlag, 1969 Zbl 0176.00801
[14] J. E. Fornæss & N. Sibony, “Complex Hénon mappings in $\mathbb{C}^2$ and Fatou-Bieberbach domains”, Duke Math. J. 65 (1992), p. 345-380
Article | Zbl 0761.32015
[15] J. E. Fornæss & N. Sibony, “Oka’s inequality for currents and applications”, Math. Ann. 301 (1995), p. 399-419
Article | Zbl 0832.32010
[16] M. Gromov, “On the entropy of holomorphic maps”, Enseignement Math. 49 (2003), p. 217-235, Manuscript (1977) Zbl 1080.37051
[17] R. Harvey & J. Polking, “Extending analytic objects”, Comm. Pure Appl. Math. 28 (1975), p. 701-727
Article | Zbl 0323.32013
[18] R. Harvey & B. Shiffman, “A characterization of holomorphic chains”, Ann. of Math. (2) 99 (1974), p. 553-587
Article | Zbl 0287.32008
[19] L. Hörmander, The analysis of Linear partial differential operators I, Springer-Verlag, 1983 Zbl 0521.35001
[20] A. Katok & B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encycl. of Math. and its Appl. 54, Cambridge Univ. Press., 1995 Zbl 0878.58020
[21] P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Dunod, 1968 Zbl 0195.11603
[22] N. Sibony, “Dynamique des applications rationnelles de $\mathbb{P}^k$”, Panoramas et Synthèses 8 (1999), p. 97-185 Zbl 1020.37026
[23] J. Smillie, “The entropy of polynomial diffeomorphisms of $\mathbb{C}^2$”, Ergodic Theory & Dynamical Systems 10 (1990), p. 823-827 Zbl 0695.58023
[24] P. Walters, An introduction to ergodic theory, Springer, 1982 Zbl 0475.28009
[25] Y. Yomdin, “Volume growth and entropy”, Israel J. Math. 57 (1987), p. 285-300
Article | Zbl 0641.54036
Tien-Cuong Dinh; Nessim Sibony
Geometry of currents, intersection theory and dynamics of horizontal-like maps
(Géométrie des courants, théorie d’intersection et dynamique des applications d’allure horizontale)
Annales de l'institut Fourier, 56 no. 2 (2006), p. 423-457, doi: 10.5802/aif.2188
Article PDF | Reviews MR 2226022 | Zbl 1089.37036
Class. Math.: 37F, 32H50, 32U40
Keywords: Structural discs of currents, Green current, equilibrium measure, mixing, entropy.
Résumé - 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.
Bibliography
Article | Zbl 0792.58034
[2] E. Bedford & J. Smillie, “Polynomial diffeomorphisms of $\mathbb{C}^2$, III: Ergodicity, exponents and entropy of the equilibrium measure”, Math. Ann. 294 (1992), p. 395-420
Article | Zbl 0765.58013
[3] J. P. Demailly, “Monge-Ampère Operators, Lelong numbers and Intersection theory in Complex Analysis and Geometry”, Plemum Press (1993), p. 115-193 Zbl 0792.32006
[4] T. C. Dinh, “Decay of correlations for Hénon maps”, to appear
arXiv
[5] T. C. Dinh, R. Dujardin & N. Sibony, “On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$”, Math. Ann. 333 (2005) no. 4, p. 703-739
Article | Zbl 1079.37040
[6] T. C. Dinh & N. Sibony, “Dynamique des applications d’allure polynomiale”, J. Math. Pures Appl. 82 (2003), p. 367-423
Article | Zbl 1033.37023
[7] T. C. Dinh & N. Sibony, “Regularization of currents and entropy”, Ann. Sci. Ecole Norm. Sup. 37 (2004), p. 959-971
Numdam | Zbl 1074.53058
[8] T. C. Dinh & N. Sibony, “Dynamics of regular birational maps in $\mathbb{P}^k$”, J. Funct. Anal. 222 (2005) no. 1, p. 202-216
Article | Zbl 1067.37055
[9] T. C. Dinh & N. Sibony, “Green currents for holomorphic automorphisms of compact Kähler manifolds”, J. Amer. Math. Soc. 18 (2005) no. 2, p. 291-312
Article | Zbl 1066.32024
[10] T. C. Dinh & N. Sibony, “Une borne supérieure pour l’entropie topologique d’une application rationnelle”, Ann. of Math. 161 (2005), p. 1637-1644
Article | Zbl 05004661
[11] R. Dujardin, “Hénon-like mappings in $\mathbb{C}^2$”, Amer. J. Math. 126 (2004), p. 439-472
Article | Zbl 1064.37035
[12] J. Duval & N. Sibony, “Polynomial convexity, rational convexity, and currents”, Duke Math. J. 79 (1995) no. 2, p. 487-513
Article | Zbl 0838.32006
[13] H. Federer, Geometric Measure Theory, Springer Verlag, 1969 Zbl 0176.00801
[14] J. E. Fornæss & N. Sibony, “Complex Hénon mappings in $\mathbb{C}^2$ and Fatou-Bieberbach domains”, Duke Math. J. 65 (1992), p. 345-380
Article | Zbl 0761.32015
[15] J. E. Fornæss & N. Sibony, “Oka’s inequality for currents and applications”, Math. Ann. 301 (1995), p. 399-419
Article | Zbl 0832.32010
[16] M. Gromov, “On the entropy of holomorphic maps”, Enseignement Math. 49 (2003), p. 217-235, Manuscript (1977) Zbl 1080.37051
[17] R. Harvey & J. Polking, “Extending analytic objects”, Comm. Pure Appl. Math. 28 (1975), p. 701-727
Article | Zbl 0323.32013
[18] R. Harvey & B. Shiffman, “A characterization of holomorphic chains”, Ann. of Math. (2) 99 (1974), p. 553-587
Article | Zbl 0287.32008
[19] L. Hörmander, The analysis of Linear partial differential operators I, Springer-Verlag, 1983 Zbl 0521.35001
[20] A. Katok & B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encycl. of Math. and its Appl. 54, Cambridge Univ. Press., 1995 Zbl 0878.58020
[21] P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Dunod, 1968 Zbl 0195.11603
[22] N. Sibony, “Dynamique des applications rationnelles de $\mathbb{P}^k$”, Panoramas et Synthèses 8 (1999), p. 97-185 Zbl 1020.37026
[23] J. Smillie, “The entropy of polynomial diffeomorphisms of $\mathbb{C}^2$”, Ergodic Theory & Dynamical Systems 10 (1990), p. 823-827 Zbl 0695.58023
[24] P. Walters, An introduction to ergodic theory, Springer, 1982 Zbl 0475.28009
[25] Y. Yomdin, “Volume growth and entropy”, Israel J. Math. 57 (1987), p. 285-300
Article | Zbl 0641.54036
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310