Geometry and dynamics of admissible metrics in measure spaces
- First Online:
- Cite this article as:
- Vershik, A.M., Zatitskiy, P.B. & Petrov, F.V. centr.eur.j.math. (2013) 11: 379. doi:10.2478/s11533-012-0149-9
- 133 Downloads
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.