Scaling entropy and automorphisms with pure point spectrum

The subject of this paper is the dynamics of metrics generated by measure-preserving transformations. Sequences of averaged metrics are considered together with the ε-entropies of the measure with respect to these metrics. The main result gives a criterion for the spectrum of a transformation to be pure point; specifically, it is shown that the scaling sequence for the ε-entropies with respect to the averages of an admissible metric is bounded if and only if the automorphism has a pure point spectrum. This paper pertains to a series of papers by the author devoted to the asymptotic theory of sequences of metric measure spaces and its applications to ergodic theory.