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Dynamics of automorphisms of \(K3\) surfaces. (Dynamique des automorphismes des surfaces \(K3\).) (French) Zbl 1045.37007
The author studies the dynamics of the automorphisms of complex surfaces \(X\) with positive topological entropy. With such an assumption, the surface is either a torus, a \(K3\) surface or a quotient of them (these are the only examples known so far). The dynamics of the tori automorphisms is well understood, as these maps are induced by an affine map of \(\mathbb {C}^2\). So one is led to the \(K3\) surface case.
The main result of the article is the following. Let \(\phi\) be an automorphism of a projective \(K3\) surface \(X\) with positive topological entropy. Then there exists a unique invariant measure with maximal entropy \(\mu\), the periodic points are equidistributed with respect to \(\mu\) and the system \((X,\mu,\phi)\) is mesurably conjugate to a Bernoulli shift. An analogous result was proved for the polynomial automorphisms of \(\mathbb {C}^2\) by E. Bedford, M. Lyubich and J. Smillie in [Invent. Math. 112, 77–125 (1993; Zbl 0792.58034)], see also the article of N. Sibony [Panor. Synth. 8, 97–185 (1999; Zbl 1020.37026)]. The proof in the \(K3\) surface case follows the same strategy. The construction of \(\mu\) is based on pluripotential theory: it is obtained as the wedge product of two dynamical closed positive currents \(T^+\) and \(T^-\) which may be introduced as limits of Nevanlinna currents associated to unstable and stable manifolds. In order to consider their product, it is crucial to prove that these currents admit continuous potential.
The author also establishes the following rigidity result. Let \(\phi\) be an automorphism of a \(K3\) surface \(X\) with infinite order. If \(\phi\) preserves two holomorphic foliations, then \(X\) is a Kummer surface and \(\phi\) comes from a linear transformation on the associated torus. Recall that a Kummer surface is the desingularization of the quotient of a torus by the involution \(-I\).

MSC:
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
14J28 \(K3\) surfaces and Enriques surfaces
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32U15 General pluripotential theory
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