Geometry and dynamics of admissible metrics in measure spaces

Anatoly Vershik 1 , Pavel Zatitskiy 1 , and Fedor Petrov 1
  • 1 St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa

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.

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