Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Zero Krengel entropy does not kill Poisson entropy

Élise Janvresse and Thierry de la Rue

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Abstract

We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann–Kakutani odometer), but whose associated Poisson suspension has positive entropy.

Résumé

Nous prouvons que les notions d’entropie de Krengel et d’entropie de Poisson pour les transformations préservant une mesure infinie ne coïncident pas toujours : nous construisons une transformation conservative préservant une mesure infinie qui a une entropie de Krengel nulle (la transformation induite sur un ensemble de mesure 1 est l’odomètre de Von Neumann–Kakutani), mais dont la suspension de Poisson a une entropie strictement positive.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 368-376.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148203

Digital Object Identifier
doi:10.1214/10-AIHP393

Mathematical Reviews number (MathSciNet)
MR2954259

Zentralblatt MATH identifier
1269.37003

Subjects
Primary: 37A05: Measure-preserving transformations 37A35: Entropy and other invariants, isomorphism, classification 37A40: Nonsingular (and infinite-measure preserving) transformations 28D20: Entropy and other invariants

Keywords
Krengel entropy Poisson suspension Infinite-measure-preserving transformation d̄-distance

Citation

Janvresse, Élise; de la Rue, Thierry. Zero Krengel entropy does not kill Poisson entropy. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 368--376. doi:10.1214/10-AIHP393. https://projecteuclid.org/euclid.aihp/1334148203


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References

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