Duke Mathematical Journal

Energy and invariant measures for birational surface maps

Eric Bedford and Jeffrey Diller

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Given a birational self-map 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.

Article information

Duke Math. J., Volume 128, Number 2 (2005), 331-368.

First available in Project Euclid: 2 June 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
Secondary: 32H50: Iteration problems 32U40: Currents


Bedford, Eric; Diller, Jeffrey. Energy and invariant measures for birational surface maps. Duke Math. J. 128 (2005), no. 2, 331--368. doi:10.1215/S0012-7094-04-12824-6. https://projecteuclid.org/euclid.dmj/1117728418

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  • W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 1984.
  • E. Bedford and J. Diller, Real and complex dynamics of a family of birational maps of the plane: The golden mean subshift, preprint.
  • E. Bedford and V. Pambuccian, Dynamics of shift-like polynomial diffeomorphisms of $ C^N$, Conform. Geom. Dyn. 2 (1998), 45--55.
  • E. Bedford and J. Smillie, Polynomial diffeomorphisms of $\mathbfC^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), 69--99.
  • E. Bedford and B. A. Taylor, Variational properties of the complex Monge-Ampère equation, I: Dirichlet principle, Duke Math. J. 45 (1978), 375--403.
  • Z. Blocki, On the definition of the Monge-Ampère operator in $\bf C^2$, Math. Ann. 328 (2004), 415--423.
  • 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.
  • N. Buchdahl, On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble) 49 (1999), 287--302.
  • S. Cantat, Dynamique des automorphismes des surfaces $K3$, Acta Math. 187 (2001), 1--57.
  • J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), 361--409.
  • --. --. --. --., ``Monge-Ampère operators, Lelong numbers and intersection theory'' in Complex Analysis and Geometry, Univ. Ser. Math, Plenum, New York, 1993, 115--193.
  • J. Diller, Dynamics of birational maps of $\mathbbP^2$, Indiana Univ. Math. J. 45 (1996), 721--772.
  • --. --. --. --., Birational maps, positive currents, and dynamics, Michigan Math. J. 46 (1999), 361--375.
  • --. --. --. --., Invariant measure and Lyapunov exponents for birational maps of $\mathbfP^2$, Comment. Math. Helv. 76 (2001), 754--780.
  • J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135--1169.
  • T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, preprint.
  • C. Favre, Points périodiques d'applications birationnelles de $\bf P^2$, Ann. Inst. Fourier (Grenoble) 48 (1998), 999--1023.
  • --. --. --. --., Note on pull-back and Lelong number of currents, Bull. Soc. Math. France 127 (1999), 445--458.
  • --. --. --. --., Multiplicity of holomorphic functions, Math. Ann. 316 (2000), 355--378.
  • C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Univ. Math. J. 50 (2001), 881--934.
  • J.-E. Fornæ ss and N. Sibony, Complex Hénon mappings in $\mathbbC^2$ and Fatou-Bieberbach domains, Duke Math. J. 65 (1992), 345--380.
  • --. --. --. --., ``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.
  • P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Lib., Wiley, New York, 1994.
  • V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math. 124 (2002), 75--106.
  • L. Hörmander, The Analysis of Linear Partial Differential Operators, I: Distribution Theory and Fourier Analysis, 2nd ed., Grundlehren Math. Wiss. 256, Springer, Berlin, 1990.
  • A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995.
  • A. Lamari, Le cône kählérien d'une surface, J. Math. Pures Appl. (9) 78 (1999), 249--263.