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Geometry and dynamics of admissible metrics in measure spaces

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Vershik A., Zatitskiy P., Petrov F.

Open Mathematics

2013

tom 11

nr 3

379-400

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.

Ograniczanie wyników

1 Open Mathematics

1 Petrov F.

1 Vershik A.

1 Zatitskiy P.

1 2013

Wyniki wyszukiwania

Geometry and dynamics of admissible metrics in measure spaces