Rank-one power weakly mixing non-singular transformations

Abstract

We show that Chacon's non-singular type III_\lambda transformation T_\lambda, 0<\lambda\leq 1, is power weakly mixing, i.e. for all sequences of non-zero integers \{k_{1},\dotsc,k_{r}\}, T_\lambda^{k_{1}}\times\dotsb\times T_\lambda^{k_{r}} is ergodic. We then show that in infinite measure, this condition is not implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.