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arxiv: 0909.1425 · v1 · submitted 2009-09-08 · 🧮 math.DS

Poisson-Pinsker factor and infinite measure preserving group actions

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keywords factorpoisson-pinskercountableentropygroupinfinitemeasurepositive
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We solve the question of the existence of a Poisson-Pinsker factor for conservative ergodic infinite measure preserving action of a countable amenable group by proving the following dichotomy: either it 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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