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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 | References | Similar Articles | Additional Information

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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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.