Flexibility of measurable and topological nilfactors in dynamical systems

TL;DR

This paper constructs minimal, uniquely ergodic dynamical systems demonstrating all possible interactions between measurable and topological nilfactors, expanding understanding of their flexibility in dynamical systems.

Contribution

It introduces new examples of minimal, uniquely ergodic systems with diverse behaviors of nilfactors, adapting Furstenberg's construction for broader applications.

Findings

Examples of systems with all possible nilfactor behaviors

Demonstrates flexibility of nilfactors in minimal systems

Adapts classical constructions to new contexts

Abstract

We construct examples of minimal and uniquely ergodic systems realizing all possible behaviors in the interplay of measurable and topological nilfactors. To build such examples, we adapt an idea that stems from Furstenberg's construction of a minimal but not uniquely ergodic system on $\mathbb{T}^2$.