Poisson-Pinsker factor and infinite measure preserving group actions

TL;DR

This paper investigates the existence of Poisson-Pinsker factors in infinite measure-preserving group actions, establishing a dichotomy based on Poisson entropy and analyzing spectral properties.

Contribution

It proves a dichotomy for conservative ergodic infinite measure actions of amenable groups regarding Poisson-Pinsker factors and entropy.

Findings

Actions either have totally positive Poisson entropy or a Poisson-Pinsker factor.

Spectral analysis shows absolute continuity in the positive entropy case.

The results specify spectral properties for abelian groups, including a9 and a9b2 cases.

Abstract

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.