Ergodic cocycles of IDPFT systems and non-singular Gaussian actions

Abstract

AbstractIt is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of non-singular infinite direct productsTof transformations$T_n$,$n\in \mathbb N$, of finite type is studied. It is shown that if$T_n$is mildly mixing,$n\in \mathbb N$, the sequence of Radon–Nikodym derivatives of$T_n$is asymptotically translation quasi-invariant andTis conservative then the Maharam extension ofTis sharply weak mixing. This technique provides a new approach to the non-singular Gaussian transformations studied recently by Arano, Isono and Marrakchi.