  We study the dynamics of the metrics generated by measure preserving
transformations. We consider a sequence of average metrics and define the
corresponding sequence of $\epsilon$-entropies ({\it scaling sequence}) of the
measure with respect to the mean metrics. The main result claims that scaling
sequences of an automorphism with respect to any {\it admissible metric} is
bounded if and only if the automorphism has discrete spectrum. This gives a
non-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 of
sequences of the metric compacts with measure and its role in dynamics.
