Conditional entropy theory in infinite measure and a question of Krengel

We develop a conditional entropy theory for infinite measure preserving actions of countable discrete amenable groups with respect to a σ-finite factor. This includes ‘infinite’ analogues of relative Kolmogorov-Sinai, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjoint...

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Bibliographic Details
Published in:Israel journal of mathematics 2009-07, Vol.172 (1), p.93-117
Main Authors: Danilenko, Alexandre I., Rudolph, Daniel J.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Entropy
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Questions
Theorems
Theoretical
Transformations
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Summary:We develop a conditional entropy theory for infinite measure preserving actions of countable discrete amenable groups with respect to a σ-finite factor. This includes ‘infinite’ analogues of relative Kolmogorov-Sinai, Rokhlin and Krieger theorems on generating partitions, Pinsker theorem on disjointness, Furstenberg decomposition and disjointness theorems, etc. In case of ℤ-action, our concept of relative entropy matches well the ‘absolute’ entropy h Kr introduced by Krengel. Answering in part his question and a question of Silva and Thieullen, we show that for any non-distal transformation S there exists an infinite measure preserving transformation T with h Kr ( T × S ) = ∞ but h Kr ( T ) = 0.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-009-0065-2