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Enveloping semigroups and quasi-discrete spectrum

Published online by Cambridge University Press:  15 June 2015

JUHO RAUTIO
JUHO RAUTIO*
Affiliation:
University of Oulu, Department of Mathematical Sciences, PL 8000, FI-90014 Oulun yliopisto, Finland email juho.rautio@oulu.fi
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Abstract

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed in 1982 by Namioka [Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, 295–300]. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on the one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalization of the property of having quasi-discrete spectrum, which is named the ${\mathcal{W}}$-property.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2627 - 2660
Copyright
© Cambridge University Press, 2015 

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