Weak mixing for nonsingular Bernoulli actions of countable amenable groups

TL;DR

This paper proves that nonsingular, conservative Bernoulli actions of countable amenable groups are weakly mixing, extending ergodic properties and answering a question posed by Z. Kosloff.

Contribution

It establishes weak mixing for nonsingular Bernoulli actions of amenable groups, a significant extension of previous ergodic results.

Findings

Nonsingular conservative Bernoulli actions are weakly mixing.

Positive answer to Z. Kosloff's question on ergodicity.

A weak pointwise ratio ergodic theorem for nonsingular actions.

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 $G$.