Geometry and dynamics of admissible metrics in measure spaces

• Autorzy: Anatoly Vershik , Pavel Zatitskiy , Fedor Petrov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Admissible metric; Measure space; Automophisms; Scaling entropy; Criteria of discreteness spectrum

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 3
• Strony: 379-400

Daty

wydano

2013-03-01

online

2012-12-22

Twórcy

autor

Anatoly Vershik

• St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa,

autor

Pavel Zatitskiy

• St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa,

autor

Fedor Petrov

• St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa,

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-012-0149-9