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Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Author: Alexandre I. Danilenko
Journal: Proc. Amer. Math. Soc. 147 (2019), 4439-4450
MSC (2010): Primary 37A20, 37A40
Published electronically: June 10, 2019
MathSciNet review: 4002554
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Abstract: Let $G$ be an amenable discrete countable infinite group, let $A$ be a finite set, and let $(\mu _g)_{g\in G}$ be a family of probability measures on $A$ such that $\inf _{g\in G}\min _{a\in A}\mu _g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes _{g\in G}(A,\mu _g)$ is nonsingular and conservative, then it is weakly mixing. This answers in the positive a question by Z. Kosloff, who proved recently that the conservative Bernoulli $\mathbb {Z}^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of $G$.

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Additional Information

Alexandre I. Danilenko
Affiliation: Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Nauky Avenue, Kharkiv, 61103, Ukraine
MR Author ID: 265198

Received by editor(s): July 16, 2018
Received by editor(s) in revised form: August 4, 2018, January 20, 2019, and January 31, 2019
Published electronically: June 10, 2019
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society