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Laminar currents and birational dynamics. (English) Zbl 1099.37037
Summary: We study the dynamics of a bimeromorphic map \(X\to X\), where \(X\) is a compact complex Kähler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic, and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of \(\mathbb{C}^2\). This extends recent results due to E. Bedford and J. Diller [Duke Math. J. 128, 331–368 (2005; Zbl 1076.37031)].

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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