Weak mixing for nonsingular Bernoulli actions of countable amenable groups

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 $G$.