Energy and invariant measures for birational surface maps

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...

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Bibliographic Details
Published in:Duke mathematical journal 2005-06, Vol.128 (2), p.331-368
Main Authors: Bedford, Eric, Diller, Jeffrey
Format: Article
Language:English
Subjects:
32A20
32H02
32H04
32H50
32U40
37F10
Currents
entire and meromorphic functions [See also 32A10
Iteration problems
Polynomials
rational maps
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Summary: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.
ISSN:0012-7094
1547-7398
DOI:10.1215/S0012-7094-04-12824-6