On systems with quasi-discrete spectrum
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”) again has 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.