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Energy and invariant measures for birational surface maps. (English) Zbl 1076.37031
Authors’ abstract: Given a birational selfmap of a compact complex surface, it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology. Under a very weak hypothesis on the map, we show how to construct such a measure. The main point in the construction is to make sense of the wedge product of two positive, closed \((1,1)\)-currents. We are able to do this in our case because local potentials for each current have “finite energy” with respect to the other. Our methods also suffice to show that the resulting measure is mixing, does not charge curves, and has nonzero Lyapunov exponents.
Reviewer: Pei-Chu Hu (Jinan)

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
32U40 Currents
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References:
[1] W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces , Ergeb. Math. Grenzgeb. (3) 4 , Springer, Berlin, 1984. · Zbl 0718.14023
[2] E. Bedford and J. Diller, Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift , · Zbl 1083.37038
[3] E. Bedford and V. Pambuccian, Dynamics of shift-like polynomial diffeomorphisms of \( C^N\) , Conform. Geom. Dyn. 2 (1998), 45–55.
[4] E. Bedford and J. Smillie, Polynomial diffeomorphisms of \(\mathbfC^2\): Currents, equilibrium measure and hyperbolicity , Invent. Math. 103 (1991), 69–99. · Zbl 0721.58037
[5] E. Bedford and B. A. Taylor, Variational properties of the complex Monge-Ampère equation, I: Dirichlet principle , Duke Math. J. 45 (1978), 375–403. · Zbl 0401.35093
[6] Z. Blocki, On the definition of the Monge-Ampère operator in \(\mathbf C^2\) , Math. Ann. 328 (2004), 415–423. · Zbl 1060.32018
[7] J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de \(\mathbfC\mathrmP^k\) , Acta Math. 182 (1999), 143–157. · Zbl 1144.37436
[8] N. Buchdahl, On compact Kähler surfaces , Ann. Inst. Fourier (Grenoble) 49 (1999), 287–302. · Zbl 0926.32025
[9] S. Cantat, Dynamique des automorphismes des surfaces \(K3\) , Acta Math. 187 (2001), 1–57. · Zbl 1045.37007
[10] J.-P. Demailly, Regularization of closed positive currents and intersection theory , J. Algebraic Geom. 1 (1992), 361–409. · Zbl 0777.32016
[11] –. –. –. –., “Monge-Ampère operators, Lelong numbers and intersection theory” in Complex Analysis and Geometry , Univ. Ser. Math, Plenum, New York, 1993, 115–193. · Zbl 0792.32006
[12] J. Diller, Dynamics of birational maps of \(\mathbbP^2\) , Indiana Univ. Math. J. 45 (1996), 721–772. · Zbl 0874.58022
[13] –. –. –. –., Birational maps, positive currents, and dynamics , Michigan Math. J. 46 (1999), 361–375. · Zbl 1062.37506
[14] –. –. –. –., Invariant measure and Lyapunov exponents for birational maps of \(\mathbfP^2\) , Comment. Math. Helv. 76 (2001), 754–780. · Zbl 1061.32012
[15] J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces , Amer. J. Math. 123 (2001), 1135–1169. · Zbl 1112.37308
[16] T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds , · Zbl 1066.32024
[17] C. Favre, Points périodiques d’applications birationnelles de \(\mathbf P^2\) , Ann. Inst. Fourier (Grenoble) 48 (1998), 999–1023. · Zbl 0924.58083
[18] –. –. –. –., Note on pull-back and Lelong number of currents , Bull. Soc. Math. France 127 (1999), 445–458. · Zbl 0937.32005
[19] –. –. –. –., Multiplicity of holomorphic functions , Math. Ann. 316 (2000), 355–378. · Zbl 0948.32020
[20] C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs , Indiana Univ. Math. J. 50 (2001), 881–934. · Zbl 1046.37026
[21] J.-E. Fornæ ss and N. Sibony, Complex Hénon mappings in \(\mathbbC^2\) and Fatou-Bieberbach domains , Duke Math. J. 65 (1992), 345–380. · Zbl 0761.32015
[22] –. –. –. –., “Complex dynamics in higher dimension, I” in Complex Analytic Methods in Dynamical Systems (Rio de Janeiro, 1992) , Astérisque 222 , Soc. Math. France, Montrouge (1994), 201–231.
[23] P. Griffiths and J. Harris, Principles of Algebraic Geometry , Wiley Classics Lib., Wiley, New York, 1994. · Zbl 0836.14001
[24] V. Guedj, Dynamics of polynomial mappings of \(\mathbbC^2\) , Amer. J. Math. 124 (2002), 75–106. · Zbl 1198.32007
[25] L. Hörmander, The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis , 2nd ed., Grundlehren Math. Wiss. 256 , Springer, Berlin, 1990. · Zbl 0712.35001
[26] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems , Encyclopedia Math. Appl. 54 , Cambridge Univ. Press, Cambridge, 1995. · Zbl 0878.58020
[27] A. Lamari, Le cône kählérien d’une surface , J. Math. Pures Appl. (9) 78 (1999), 249–263. · Zbl 0941.32007
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