Top-nilpotent enveloping semigroups and pro-nilsystems

TL;DR

This paper establishes a precise characterization of minimal systems as d-step pro-nilsystems based on the top-nilpotency of their enveloping semigroups, resolving an open question in the field.

Contribution

It proves that a minimal system is a d-step pro-nilsystem if and only if its enveloping semigroup is a d-step top-nilpotent group, linking algebraic and dynamical properties.

Findings

Characterization of minimal systems via top-nilpotent enveloping semigroups

Resolution of an open question by Donoso

Equivalence between pro-nilsystems and top-nilpotent enveloping semigroups

Abstract

In this paper, it is shown that for $d\in\mathbb{N}$, a minimal system $(X,T)$ is a $d$-step pro-nilsystem if its enveloping semigroup is a $d$-step top-nilpotent group, answering an open question by Donoso. Thus, combining the previous result of Donoso, it turns out that a minimal system $(X,T)$ is a $d$-step pro-nilsystem if and only if its enveloping semigroup is a $d$-step top-nilpotent group.