Geometry and dynamics of admissible metrics in measure spaces

Anatoly Vershik; Pavel Zatitskiy; Fedor Petrov

Open Mathematics (2013)

  • Volume: 11, Issue: 3, page 379-400
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Anatoly Vershik, Pavel Zatitskiy, and Fedor Petrov. "Geometry and dynamics of admissible metrics in measure spaces." Open Mathematics 11.3 (2013): 379-400. <http://eudml.org/doc/269752>.

@article{AnatolyVershik2013,
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.},
author = {Anatoly Vershik, Pavel Zatitskiy, Fedor Petrov},
journal = {Open Mathematics},
keywords = {Admissible metric; Measure space; Automophisms; Scaling entropy; Criteria of discreteness spectrum; admissible metric; measure space; measure-preserving transformation; scaling entropy; discrete spectrum},
language = {eng},
number = {3},
pages = {379-400},
title = {Geometry and dynamics of admissible metrics in measure spaces},
url = {http://eudml.org/doc/269752},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Anatoly Vershik
AU - Pavel Zatitskiy
AU - Fedor Petrov
TI - Geometry and dynamics of admissible metrics in measure spaces
JO - Open Mathematics
PY - 2013
VL - 11
IS - 3
SP - 379
EP - 400
AB - 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.
LA - eng
KW - Admissible metric; Measure space; Automophisms; Scaling entropy; Criteria of discreteness spectrum; admissible metric; measure space; measure-preserving transformation; scaling entropy; discrete spectrum
UR - http://eudml.org/doc/269752
ER -

References

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