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arXiv · 1205.1174

Geometry and Dynamics of Admissible Metrics in Measure Spaces

A. Vershik·F. Petrov·P. Zatitskiy
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Abstract

We study a wide class of metrics in a Lebesgue space with a standard measure, 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 "-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 "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded.

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A. Vershik, F. Petrov, P. Zatitskiy. 2012-10-25. Geometry and Dynamics of Admissible Metrics in Measure Spaces. https://arxiv.org/abs/1205.1174

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