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Scaling entropy and automorphisms with pure point spectrum

Author: A. M. Vershik
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 1.
Journal: St. Petersburg Math. J. 23 (2012), 75-91
MSC (2010): Primary 37A35
Published electronically: November 8, 2011
MathSciNet review: 2760149
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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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Additional Information

A. M. Vershik
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, Petersburg 191023, Russia

Keywords: Admissible metric, scaling entropy, pure point spectrum
Received by editor(s): September 15, 2010
Published electronically: November 8, 2011
Additional Notes: Partially supported by RFBR (grants nos. RFBR-08-01-00379-a and RFBR-09-01-12175-ofi-m).
Dedicated: To the memory of my friend Misha Birman
Article copyright: © Copyright 2011 American Mathematical Society