Cube structures of the universal minimal system, nilsystems and applications

TL;DR

This paper introduces a new approach using cube structures to analyze nilsystems and their extensions, providing alternative proofs and new results in the structural theory of topological systems.

Contribution

It develops a novel cube-structure-based method to study nilsystems, offering new proofs and insights, including an algebraic description of a key equivalence relation.

Findings

Proved that $ extbf{RP}^{[d]}$ is an equivalence relation.

Provided alternative proofs for saturation properties of factor maps.

Established new applications to the structural theory of topological systems.

Abstract

We propose and develop an approach to study nilsystems and their proximal extensions using cube structures associated with the universal minimal system. We provide alternative proofs for results regarding saturation properties of factor maps to maximal nilfactors in cubes, as well as new results and applications of independent interest to the structural theory of topological systems. In particular, we give a new proof that $\mathbf{RP}^{[d]}$ is an equivalence relation, building upon the distal case, by establishing a description of this relation in algebraic terms. This is new even for d=1.