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We study the dynamics of the metrics generated by measure preservingtransformations. We consider a sequence of average metrics and define thecorresponding sequence of $\epsilon$-entropies ({\it scaling sequence}) of themeasure with respect to the mean metrics. The main result claims that scalingsequences of an automorphism with respect to any {\it admissible metric} isbounded if and only if the automorphism has discrete spectrum. This gives anon-spectral criterion of the discreteness of the spectrum of an automorphism.The related result was discussed in \cite{Fe} but our approach is different.This article is one in the series of papers about asymptotic theory ofsequences of the metric compacts with measure and its role in dynamics.

Scaling entropy and automorphisms with purely point spectrum