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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Amenable actions and almost invariant sets


Authors: Alexander S. Kechris and Todor Tsankov
Journal: Proc. Amer. Math. Soc. 136 (2008), 687-697
MSC (2000): Primary 28D15; Secondary 43A07
Published electronically: November 3, 2007
MathSciNet review: 2358510
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Abstract: In this paper, we study the connections between properties of the action of a countable group $ \Gamma$ on a countable set $ X$ and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of $ \Gamma$ on $ M^X$, where $ M$ is a measure space. In particular, we show that the action of $ \Gamma$ on $ X$ is amenable iff the shift $ \Gamma \curvearrowright M^X$ has almost invariant sets.


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Additional Information

Alexander S. Kechris
Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
Email: kechris@caltech.edu

Todor Tsankov
Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
Email: todor@caltech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09116-2
PII: S 0002-9939(07)09116-2
Keywords: Generalized Bernoulli shifts, amenable actions, almost invariant sets, $E_0$-ergodicity
Received by editor(s): October 2, 2006
Received by editor(s) in revised form: February 8, 2007
Published electronically: November 3, 2007
Additional Notes: This research was partially supported by NSF grant DMS-0455285
Communicated by: Jane M. Hawkins
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.