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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0149-9,
author = {Anatoly Vershik and Pavel Zatitskiy and Fedor Petrov},
title = {Geometry and dynamics of admissible metrics in measure spaces},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {379-400},
zbl = {1261.37004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0149-9}
}
Anatoly Vershik; Pavel Zatitskiy; Fedor Petrov. Geometry and dynamics of admissible metrics in measure spaces. Open Mathematics, Tome 11 (2013) pp. 379-400. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0149-9/
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