Enveloping semigroups of systems of order d

TL;DR

This paper investigates the structure of enveloping semigroups in d-step nilsystems, demonstrating their topological nilpotency and establishing equivalences for the case d=2, thereby extending existing mathematical results.

Contribution

It proves that d-step nilsystems have d-step topologically nilpotent enveloping semigroups and shows the equivalence of certain notions when d=2, extending prior work.

Findings

d-step nilsystems have d-step topologically nilpotent enveloping semigroups

For d=2, certain properties are equivalent, extending previous results

Uses cube machinery by Host, Kra, and Maass

Abstract

In this paper we study the Ellis semigroup of a d-step nilsystem and the inverse limit of such systems. By using the machinery of cubes developed by Host, Kra and Maass, we prove that such a system has a d-step topologically nilpotent enveloping semigroup. In the case d=2, we prove that these notions are equivalent, extending a previous result by Glasner.