On systems with quasi-discrete spectrum

TL;DR

This paper revisits the theory of systems with quasi-discrete spectrum, providing simplified proofs of key theorems, classifying factors algebraically, and addressing open questions about Markov quasi-factors under certain conditions.

Contribution

It offers simplified proofs of classical theorems, provides a complete algebraic classification of factors, and advances understanding of Markov quasi-factors in quasi-discrete spectrum systems.

Findings

Simplified proofs of Hahn--Parry and Abramov theorems.

Complete algebraic classification of factors of QDS-systems.

Proved that under certain conditions, Markov quasi-factors are actual factors.

Abstract

In this paper we re-examine the theory of systems with quasi-discrete spectrum initiated in the 1960's by Abramov, Hahn, and Parry. In the first part, we give a simpler proof of the Hahn--Parry theorem stating that each minimal topological system with quasi-discrete spectrum is isomorphic to a certain affine automorphism system on some compact Abelian group. Next, we show that a suitable application of Gelfand's theorem renders Abramov's theorem --- the analogue of the Hahn-Parry theorem for measure-preserving systems --- a straightforward corollary of the Hahn-Parry result. In the second part, independent of the first, we present a shortened proof of the fact that each factor of a totally ergodic system with quasi-discrete spectrum (a "QDS-system") has again quasi-discrete spectrum and that such systems have zero entropy. Moreover, we obtain a complete algebraic classification of the factors of a QDS-system. In the third part, we apply the results of the second to the (still open) question whether a Markov quasi-factor of a QDS-system is already a factor of it. We show that this is true when the system satisfies some algebraic constraint on the group of quasi-eigenvalues, which is satisfied, e.g., in the case of the skew shift.