Scaling entropy and automorphisms with pure point spectrum
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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.References
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Bibliographic Information
- A. M. Vershik
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, Petersburg 191023, Russia
- MR Author ID: 178105
- Email: vershik@pdmi.ras.ru
- 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).
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 75-91
- MSC (2010): Primary 37A35
- DOI: https://doi.org/10.1090/S1061-0022-2011-01187-2
- MathSciNet review: 2760149
Dedicated: To the memory of my friend Misha Birman