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Central European Journal of Mathematics

, Volume 11, Issue 3, pp 379–400 | Cite as

Geometry and dynamics of admissible metrics in measure spaces

  • Anatoly M. Vershik
  • Pavel B. Zatitskiy
  • Fedor V. Petrov
Research Article
  • 134 Downloads

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.

Keywords

Admissible metric Measure space Automophisms Scaling entropy Criteria of discreteness spectrum 

MSC

37A05 11J83 37C85 

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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  • Anatoly M. Vershik
    • 1
  • Pavel B. Zatitskiy
    • 1
  • Fedor V. Petrov
    • 1
  1. 1.St. Petersbrug Branch of Mathematical Institute of Russian Academy of ScienceSt. PetersbrugRussa

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