Poisson-Pinsker factor and infinite measure preserving group actions

Roy, Emmanuel

doi:10.1090/S0002-9939-10-10224-X

Poisson-Pinsker factor and infinite measure preserving group actions
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Proc. Amer. Math. Soc. 138 (2010), 2087-2094 Request permission

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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