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

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. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-Ziegler structure theorem is a key input in the companion paper by the second author and Fraczyk classifying point processes (i.e. random subsets) of $\mathbb{Z}^d$ whose law is invariant under the group $\mathrm{ASL}_d(\mathbb{Z})$ of affine transformations.