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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 22, 2012

Geometry and dynamics of admissible metrics in measure spaces

  • Anatoly Vershik EMAIL logo , Pavel Zatitskiy and Fedor Petrov
From the journal Open Mathematics

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.

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Published Online: 2012-12-22
Published in Print: 2013-3-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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