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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
##### Keywords:
birational selfmap; energy; invariant measure; currents; mixing
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