Scaling entropy and automorphisms with purely point spectrum

Details Scaling entropy and automorphisms with purely point spectrum Scaling entropy and automorphisms with purely point spectrum

2010-08-29

arxiv.org/abs/1008.4946v4

Mathematics

Dynamical Systems

27 pp.28 Ref; St.Petersburg Mathematical Journal, No.1, 2011

Scientific paper

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.

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