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Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy. (English) Zbl 1476.37071

Summary: We classify complex projective surfaces \(X\) with an automorphism \(f\) of positive entropy for which the unique measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a byproduct, if \(X\) is a \(K3\) surface and is not a Kummer surface, the periodic points of \(f\) are equidistributed with respect to a probability measure which is singular with respect to the canonical volume of \(X\). The proof is based on complex algebraic geometry and Hodge theory, Pesin’s theory and renormalization techniques. A crucial argument relies on a new compactness property of entire curves parametrizing the invariant manifolds of the automorphism.

MSC:

37F80 Higher-dimensional holomorphic and meromorphic dynamics
14J50 Automorphisms of surfaces and higher-dimensional varieties
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32U40 Currents
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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