Ergodic cocycles of IDPFT systems and nonsingular Gaussian actions

TL;DR

This paper investigates the ergodic properties of Gaussian cocycles over certain transformations, showing they are either coboundaries or weakly mixing, and explores the Maharam extension of infinite product transformations for new insights.

Contribution

It introduces a novel approach to analyzing nonsingular Gaussian transformations and characterizes their ergodic behavior in terms of cocycles and Maharam extensions.

Findings

Gaussian cocycles are either coboundaries or sharply weak mixing.

Infinite direct product transformations with mildly mixing components have sharply weak mixing Maharam extensions.

New techniques connect Gaussian cocycle properties with nonsingular transformation ergodicity.

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.