Rank-one power weakly mixing non-singular transformations (Q2757006)

An infinite measure preserving transformation \(T\) has infinite ergodic index if for all \(k>0\), the Cartesian products \(T\times\ldots\times T\) are ergodic. Generally \(T\times T\) ergodic does not imply \(T\) of infinite ergodic index.NEWLINENEWLINENEWLINEThe transformation \(T\) is power weakly mixing if for all finite sequences of non-zero integers \(\{k_{1},\ldots, k_{r}\}\), \(T^{k_{1}}\times\ldots \times T^{k_{r}}\) is ergodic.NEWLINENEWLINENEWLINEIn the paper under review it is shown that infinite ergodic index does neither imply power weak mixing nor \(2\)-recurrence. The result is proved for Chacon's non-singular type III transformation \(T_{\lambda}\), \(0<\lambda\leq 1\).