Geometry and dynamics of admissible metrics in measure spaces

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 e{open}-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 e{open}-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded. © 2013 Versita Warsaw and Springer-Verlag Wien.