Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy

Abstract

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.