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

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