Entropy and flatness in local algebraic dynamics

Abstract

For a local endomorphism of a noetherian local ring we introduce a notion of entropy, along with two other asymptotic invariants. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that everyfinite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to afinite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a pointfixed by afinite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the fine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.