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. http://projecteuclid.org/euclid.dmj/1117728418.

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