9 research outputs found
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
On systems with quasi-discrete spectrum
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.Comment: 25 pages. Accepted for publication in Studia Mathematic