Scaling entropy and automorphisms with purely point spectrum∗

We study the dynamics of metrics generated by measure-preserving transformations. We consider sequences of average metrics and entropies of the measure with respect to these metrics. The main result, which gives a criterion for checking that the spectrum of a transformation if purely point, is that the scaling sequence for the -entropies with respect to the averages of an admissible metric is bounded if and only if the automorphism has a purely point spectrum. This paper is one of a series of papers by the author devoted to the asymptotic theory of sequences of metric measure spaces and its applications to ergodic theory.