Conditional entropy theory in infinite measure and a question of Krengel
type: Item
instance of: scholarly article
title: Conditional entropy theory in infinite measure and a question of Krengel
zbMATH Open document ID: 1177.28038
author: Alexandre I. Danilenko
author: Daniel J. Rudolph
published in: Israel Journal of Mathematics
publication date: 2009-10-02T00:00:00Z
review text: Let \(T\) be an infinite measure preserving transformation. In the paper it is introduced in a natural way a conditional entropy \(h(T,{\mathfrak F})\) of \(T\) relative to a \(\sigma\)-finite factor \({\mathfrak F}\). To this end it is applied the orbital approach to conditional entropy. It is shown that \(h(T|{\mathfrak F})\) is a factor orbit invariant as in the finite measure preserving case. This means that the infinite relative entropy, as in finite measure case, depends only on the \((T\uparrow {\mathfrak F})\)-orbit equivalence relation \(\mathcal R\) and the extending (Rokhlin) cocycle of \(\mathcal R\). It appears that the infinite conditional entropy theory is similar (but not identical) to the classical probability case. There are infinite analogues of relative Kolmogorov-Sinai entropy, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition disjointness theorem, etc. In case of \(\mathbb Z\)-action, the concept of relative entropy matches well the entropy \(h_Kr \) introduced by Krengel. Ansering his question and a question of Silva and Thieullen, it is shown that for any non-distal transformation \(S\) there exists an infinite measure preserving transformation \(T\) with \(h_Kr (T\times S)=\infty\) but \(h_kr (T)=0\).
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37A35
zbMATH DE Number: 5610599
zbMATH Keywords: infinite measure preserving transformations
zbMATH Keywords: conditional entropy
zbMATH Keywords: disjointness
zbMATH Keywords: amenable groups
MaRDI profile type: Publication
full work available at URL: s11856-009-0065-2
OpenAlex ID: W2080983306
cites work: Amenable groups, stationary measures and partitions with independent iterates
cites work: Partitions with independent iterates in random dynamical systems
cites work: An amenable equivalence relation is generated by a single transformation
cites work: Entropy theory from the orbital point of view
cites work: Funny rank-one weak mixing for nonsingular Abelian actions
cites work: Q5459668
cites work: Generators and Bernoullian factors for amenable actions and cocycles on their orbits
cites work: Q5711910
cites work: The spectrum of completely positive entropy actions of countable amenable groups
cites work: Non-Bernoulli systems with completely positive entropy
cites work: Q3909268
cites work: Entropy theory without a past
cites work: Quasi-Factors of Zero Entropy Systems
cites work: A simple map with no prime factors
cites work: Q4321894
cites work: Sequence Entropy Relative to an Invariant σ-Algebra
cites work: On the entropy of a product endomorphism in infinite measure spaces
cites work: Entropy of conservative transformations
cites work: Q5599279
cites work: Average co-ordinate entropy
cites work: An invariant for non-singular isomorphism
cites work: Entropy and isomorphism theorems for actions of amenable groups
cites work: An Ergodic Theorem of Information Theory Without Invariant Measure
cites work: Ergodic and Spectral Analysis of Certain Infinite Measure Preserving Transformations
cites work: Q5565255
cites work: Entropy and mixing for amenable group actions
cites work: LECTURES ON THE ENTROPY THEORY OF MEASURE-PRESERVING TRANSFORMATIONS
cites work: A Skew Product Entropy for Nonsingular Transformations
cites work: Dynamic theory of growth in groups: Entropy, boundaries, examples
cites work: Ergodic actions with generalized discrete spectrum
DOI: 10.1007/S11856-009-0065-2
description: scientific article; zbMATH DE number 5610599