Strictly ergodic distal models and a new approach to the Host-Kra factors

TL;DR

This paper introduces nilcycles as measurable counterparts to cocycles, providing new conditions for ergodic group extensions of distal systems to have strictly ergodic models, and offers a novel proof that Host-Kra factors are inverse limits of nilsystems.

Contribution

It develops the concept of nilcycles, enabling new criteria for modeling ergodic extensions and offers a new proof of the structure of Host-Kra factors as inverse limits of nilsystems.

Findings

Nilcycles facilitate modeling of ergodic group extensions.

Conditions established for strictly ergodic distal models.

New proof that Host-Kra factors are inverse limits of nilsystems.

Abstract

Cocycles are a key object in Antol\'{i}n Camarena and Szegedy's (topological) theory of nilspaces. We introduce measurable counterparts, named nilcycles, enabling us to give conditions which guarantee that an ergodic group extension of a strictly ergodic distal system admits a strictly ergodic distal topological model, revisiting a problem studied by Lindenstrauss. In particular we show that if the base space is a dynamical nilspace then a dynamical nilspace topological model may be chosen for the extension. This approach combined with a structure theorem of Gutman, Manners and Varj\'{u} applied to the ergodic group extensions between successive Host-Kra characteristic factors gives a new proof that these factors are inverse limit of nilsystems.