Zero Krengel entropy does not kill Poisson entropy

Élise Janvresse; Thierry de la Rue

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 368-376
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Janvresse, Élise, and de la Rue, Thierry. "Zero Krengel entropy does not kill Poisson entropy." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 368-376. <http://eudml.org/doc/272088>.

@article{Janvresse2012,
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.},
author = {Janvresse, Élise, de la Rue, Thierry},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Krengel entropy; Poisson suspension; infinite-measure-preserving transformation; d̄-distance; infinite measure-preserving transformation; -distance},
language = {eng},
number = {2},
pages = {368-376},
publisher = {Gauthier-Villars},
title = {Zero Krengel entropy does not kill Poisson entropy},
url = {http://eudml.org/doc/272088},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Janvresse, Élise
AU - de la Rue, Thierry
TI - Zero Krengel entropy does not kill Poisson entropy
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 368
EP - 376
AB - 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.
LA - eng
KW - Krengel entropy; Poisson suspension; infinite-measure-preserving transformation; d̄-distance; infinite measure-preserving transformation; -distance
UR - http://eudml.org/doc/272088
ER -

References

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  1. [1] J. Aaronson and K. K. Park. Predictability, entropy and information of infinite transformations. Fund. Math.206 (2009) 1–21. Zbl1187.37014MR2576257
  2. [2] P. Billingsley. Probability and Measure, 2nd edition. Wiley, New York, 1986. Zbl0411.60001MR830424
  3. [3] É. Janvresse, T. Meyerovitch, E. Roy and T. de la Rue. Poisson suspensions and entropy for infinite transformations. Trans. Amer. Math. Soc.362 (2010) 3069–3094. Zbl1196.37015MR2592946
  4. [4] U. Krengel. Entropy of conservative transformations. Z. Wahrsch. Verw. Gebiete7 (1967) 161–181. Zbl0183.19303MR218522
  5. [5] U. Krengel. On certain analogous difficulties in the investigation of flows in a probability space and of transformations in an infinite measure space. In Functional Analysis (Proc. Sympos., Monterey, CA, 1969) 75–91. Academic Press, New York, 1969. Zbl0268.28010MR265558
  6. [6] D. S. Ornstein. Ergodic Theory, Randomness, and Dynamical Systems. Yale Univ. Press, New Haven, CT, 1974. Zbl0296.28016MR447525
  7. [7] W. Parry. Entropy and Generators in Ergodic Theory. W. A. Benjamin, New York, 1969. Zbl0175.34001MR262464
  8. [8] E. Roy. Mesures de poisson, infinie divisibilité et propriétés ergodiques. Ph.D. thesis, 2005. 
  9. [9] P. C. Shields. The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics 13. Amer. Math. Soc. Providence, RI, 1996. Zbl0879.28031MR1400225

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