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Poisson-Pinsker factor and infinite measure preserving group actions

Author: Emmanuel Roy
Journal: Proc. Amer. Math. Soc. 138 (2010), 2087-2094
MSC (2010): Primary 37A40, 37A35, 60G51; Secondary 37A15, 37A50
Published electronically: February 5, 2010
MathSciNet review: 2596046
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Abstract: We solve the problem of the existence of a Poisson-Pinsker factor for a conservative ergodic infinite measure preserving action of a countable amenable group by proving the following dichotomy: either the system has totally positive Poisson entropy (and is of zero type) or it possesses a Poisson-Pinsker factor. If $ G$ is abelian and the entropy positive, the spectrum is absolutely continuous (Lebesgue countable if $ G=\mathbb{Z}$) on the whole $ L^{2}$-space in the first case and in the orthocomplement of the $ L^{2}$-space of the Poisson-Pinsker factor in the second.

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  • 1. J. Aaronson and K. K. Park, Predictability, entropy and information of infinite transformations, Fund. Math. 206 (2009), 1-21.
  • 2. I. A. Danilenko and D. J. Rudolph, Conditional entropy theory in infinite measure and a question of Krengel, Israel J. Math. 172 (2009), no. 1, 93-117.
  • 3. T. de la Rue and E. Janvresse, Krengel entropy does not kill Poisson entropy, preprint.
  • 4. Y. Derriennic, K. Fraczek, M. Lema $ \acute{\mathrm{n}}$czyk, and F. Parreau, Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings, Colloq. Math. 110 (2008), 81-115. MR 2353900 (2008j:37001)
  • 5. A. H. Dooley and V. Ya. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, J. Funct. Anal. 196 (2002), 1-18. MR 1941988 (2003m:37006)
  • 6. E. Glasner, J.-P. Thouvenot, and B. Weiss, Entropy theory without a past, Ergodic Theory Dynam. Systems 20 (2000), 1355-1370. MR 1786718 (2001h:37011)
  • 7. E. Janvresse, T. Meyerovitch, T. de la Rue, and E. Roy, Poisson suspensions and entropy for infinite transformations, Trans. Amer. Math. Soc., to appear.
  • 8. U. Krengel, Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Geb. 7 (1967), 161-181. MR 0218522 (36:1608)
  • 9. W. Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0262464 (41:7071)
  • 10. E. Roy, Mesures de Poisson, infinie divisibilité et propriétés ergodiques, Ph.D. thesis, Université Paris 6, 2005.
  • 11. -, Poisson suspensions and infinite ergodic theory, Ergodic Theory Dynam. Systems 29 (2009), no. 2, 667-683. MR 2486789
  • 12. J.-P. Thouvenot, Une classe de systèmes pour lesquels la conjecture de Pinsker est vraie, Israel J. Math. 21 (1975), no. 2-3, 208-214. MR 0382602 (52:3484)

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Additional Information

Emmanuel Roy
Affiliation: Laboratoire Analyse Géométrie et Applications, UMR 7539, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France

Keywords: Poisson suspensions, infinite ergodic theory, joinings
Received by editor(s): March 29, 2009
Received by editor(s) in revised form: September 22, 2009
Published electronically: February 5, 2010
Additional Notes: This paper was written during the MSRI semester program “Ergodic Theory and Additive Combinatorics” in Berkeley. The author is very grateful to this institution and to the organizers of this program for funding his research during this semester.
Communicated by: Bryna Kra
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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