Scaling entropy and automorphisms with pure point spectrum

Vershik, A.

doi:10.1090/S1061-0022-2011-01187-2

Scaling entropy and automorphisms with pure point spectrum
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by A. M. Vershik
Translated by: the author

St. Petersburg Math. J. 23 (2012), 75-91

DOI: https://doi.org/10.1090/S1061-0022-2011-01187-2

Published electronically: November 8, 2011

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Abstract:

The subject of this paper is the dynamics of metrics generated by measure-preserving transformations. Sequences of averaged metrics are considered together with the $\epsilon$-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 $\epsilon$-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.

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