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arXiv · 1807.05905

Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Alexandre I. Danilenko
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Abstract

Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(\mu_g)_{g\in G}$ 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 positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-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. 2018-07-16. Weak mixing for nonsingular Bernoulli actions of countable amenable groups. https://arxiv.org/abs/1807.05905

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