Geometry and dynamics of admissible metrics in measure spaces

By Anatoly M. Vershik, P. Zatitskiy, F. Petrov. 2012

54 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2b19509f285f79479ee6c19d1502894f21d9ed47

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Jian Li
2. Xiangdong Ye
3. Wen Huang
4. Leiye Xu
5. J. Thouvenot
6. Eli Glasner
7. Felipe Garc'ia-Ramos
8. Guohua Zhang

In Mathematical Physics

1. Jian Li
2. Xiangdong Ye
3. Wen Huang
4. J. Thouvenot
5. Leiye Xu
6. Eli Glasner
7. Felipe Garc'ia-Ramos
8. Guohua Zhang

Citations in the corpus

• Bounded complexity, mean equicontinuity and discrete spectrum (2018, methodology): “It should be noticed that in the introduction of [32] it requires a mild condition that the standard (Lebesgue) space (X, μ) is non-atomic.”