Back to Search
Start Over
Geometry and dynamics of admissible metrics in measure spaces
- Source :
- Open Mathematics, Vol 11, Iss 3, Pp 379-400 (2013)
- Publication Year :
- 2013
- Publisher :
- De Gruyter, 2013.
-
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.<br />Comment: 37p. Ref.19
- Subjects :
- 37c85
General Mathematics
Geometry
Dynamical Systems (math.DS)
37a05
measure space
FOS: Mathematics
admissible metric
QA1-939
Uniform boundedness
Ergodic theory
Mathematics - Dynamical Systems
Mathematics
Discrete mathematics
11j83
criteria of discreteness spectrum
scaling entropy
Equivalence of metrics
28D20, 37A35, 54E35
Number theory
Compact space
Norm (mathematics)
Standard probability space
Invariant measure
automophisms
Subjects
Details
- Language :
- English
- ISSN :
- 23915455
- Volume :
- 11
- Issue :
- 3
- Database :
- OpenAIRE
- Journal :
- Open Mathematics
- Accession number :
- edsair.doi.dedup.....898a43434729762e90a00fc7f41d2d2f