2015/09/29 by Markus Haase, Haase, Markus, Nikita Moriakov +1
Mathematics · #22Cxx #37A35 #37B05 #47A35 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #math.DS #math.FA #msc:22Cxx #msc:37A35 #msc:37B05 #msc:47A35
paper · pdf · doi:10.48550/arxiv.1509.08961
25 pages. Accepted for publication in Studia Mathematica
arxiv created 2017/06/01 · arxiv updated 2017/06/02
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.