The non-ergodic Host-Kra-Ziegler structure theorem for $\mathbb{Z}^d$-actions via measurable selections

TL;DR

This paper extends the Host-Kra-Ziegler structure theorem to non-ergodic measure-preserving actions of bd, using a measurable selection approach to reduce to the ergodic case, advancing understanding of bd-dynamics.

Contribution

It introduces a non-ergodic version of the structure theorem for bd-actions, employing a measurable selection method to connect with the ergodic case.

Findings

Established a non-ergodic structure theorem for bd-actions.

Reduced non-ergodic case to ergodic case using measurable selections.

Extended the applicability of the Host-Kra-Ziegler theorem.

Abstract

We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\ge 2$ due to Candela and Szegedy) via a measurable selection procedure.