Ergodic cocycles of IDPFT systems and nonsingular Gaussian actions

Abstract

It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of nonsingular infinite direct products $T$ of transformations $T_n$, $n\in\Bbb N$, of finite type (IDPFT) is studied. It is shown that if $T_n$ is mildly mixing, $n\in\Bbb N$, the sequence of the Radon-Nikodym derivatives of $T_n$ is asymptotically translation quasi-invariant and $T$ is conservative then the Maharam extension of $T$ is sharply weak mixing. This techniques provides a new approach to the nonsingular Gaussian transformations studied recently by Arano, Isono and Marrakchi.