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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 11, Issue 3

Issues

Volume 13 (2015)

Geometry and dynamics of admissible metrics in measure spaces

Anatoly Vershik
  • St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa
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/ Pavel Zatitskiy
  • St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa
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/ Fedor Petrov
  • St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa
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Published Online: 2012-12-22 | DOI: https://doi.org/10.2478/s11533-012-0149-9

Abstract

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

MSC: 37A05; 11J83; 37C85

Keywords: Admissible metric; Measure space; Automophisms; Scaling entropy; Criteria of discreteness spectrum

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About the article

Published Online: 2012-12-22

Published in Print: 2013-03-01


Citation Information: Open Mathematics, Volume 11, Issue 3, Pages 379–400, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-012-0149-9.

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© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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