Topological characteristic factors along cubes of minimal systems

TL;DR

This paper investigates the topological characteristic factors along cubes in minimal systems, establishing that pro-nilfactors serve as these factors up to proximal extensions, especially in distal minimal systems.

Contribution

It demonstrates that pro-nilfactors are the topological characteristic factors along cubes of minimal systems, providing a detailed structural understanding.

Findings

Pro-nilfactors are the topological characteristic factors along cubes.

Maximal (d-1)-step pro-nilfactors are the cubic characteristic factors in distal minimal systems.

The results hold up to proximal extensions.

Abstract

In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal $(d-1)$-step pro-nilfactor is the topological cubic characteristic factor of order $d$.