Enveloping semigroups and quasi-discrete spectrum

TL;DR

This paper characterizes the structure of enveloping semigroups in certain distal dynamical systems, solves open problems from 1982, and introduces the universal minimal system with quasi-discrete spectrum, expanding understanding of spectral properties.

Contribution

It provides a detailed description of enveloping semigroups for elementary systems, constructs the universal minimal system with quasi-discrete spectrum, and introduces the w-property as a generalization.

Findings

Enveloping semigroups of elementary systems are characterized.

The universal minimal system with quasi-discrete spectrum is constructed.

A minimal system is a factor of this universal system iff its enveloping semigroup has quasi-discrete spectrum.

Abstract

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed by Namioka in 1982. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalisation of the property of having quasi-discrete spectrum, which is named the $\mathcal{W}$-property.