### Abstract

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 quasidiscrete 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.

Original language | English |
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Pages (from-to) | 173-199 |

Number of pages | 27 |

Journal | Studia Mathematica |

Volume | 241 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2018 |

### Keywords

- Isomorphism theorem
- Markov quasifactor
- Quasi-discrete spectrum

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## Cite this

*Studia Mathematica*,

*241*(2), 173-199. https://doi.org/10.4064/sm8756-6-2017