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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Poisson suspensions and entropy for infinite transformations


Authors: Élise Janvresse, Tom Meyerovitch, Emmanuel Roy and Thierry de la Rue
Journal: Trans. Amer. Math. Soc. 362 (2010), 3069-3094
MSC (2000): Primary 37A05, 37A35, 37A40, 28D20
Published electronically: December 17, 2009
MathSciNet review: 2592946
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Abstract: The Poisson entropy of an infinite-measure-preserving transformation is defined in the 2005 thesis of Roy as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel's and Parry's entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy, $ -\sum q_i p_{i,j}\log p_{i,j}$, holds for any definitions of entropy. Poisson entropy dominates Parry's entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel's entropies equals difference of the Poisson entropies. In case there already exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised by Aaronson and Park about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations.


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Additional Information

Élise Janvresse
Affiliation: Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, F76801 Saint Étienne du Rouvray, France
Email: Elise.Janvresse@univ-rouen.fr

Tom Meyerovitch
Affiliation: School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
Email: tomm@post.tau.ac.il

Emmanuel Roy
Affiliation: Laboratoire Analyse, Géométrie et Applications, Université Paris 13 Institut Galilée, 99 avenue Jean-Baptiste Clément, F93430 Villetaneuse, France
Email: roy@math.univ-paris13.fr

Thierry de la Rue
Affiliation: Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, CNRS, Avenue de l’Université, F76801 Saint Étienne du Rouvray, France
Email: Thierry.de-la-Rue@univ-rouen.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04968-X
PII: S 0002-9947(09)04968-X
Received by editor(s): March 25, 2008
Published electronically: December 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.