Infinite rank one actions and nonsingular Chacon transformations

Abstract

Let $G$ be a discrete countable Abelian group. We construct an infinite measure preserving rank one action $T=(T_g)$ of $G$ such that (i) the transformation $T_g$ has infinite ergodic index but $T_g\times T_{2g}$ is not ergodic for any element $g$ of infinite order, (ii) $T_{g_1}\times\cdots\times T_{g_n}$ is conservative for every finite sequence $g_1,\dots, g_n\in G$. In the case $G=\mathbb{Z}$ this answers a question of C. Silva. Moreover, we show that

(i) every weakly stationary nonsingular Chacon transformation with 2-cuts is power weakly mixing and

(ii) every weakly stationary nonsingular Chacon$^*$ transformation with 2-cuts has infinite ergodic index but is not power weakly mixing.