Topology Atlas Document # ppae-11


The Cantor set of linear orders on N is the universal minimal S\infty-system

Eli Glasner

Proceedings of the Ninth Prague Topological Symposium (2001) pp. 119-123

Each topological group G admits a unique universal minimal dynamical system (M(G), G). For a locally compact non-compact group this is a nonmetrizable system with a very rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. One such group is the topological group S of all the permutations of the integers Z, with the topology of pointwise convergence. In this paper we show that (M(S), S) is a symbolic dynamical system (hence in particular M(S) is a Cantor set), and we give a full description of all its symbolic factors.

Mathematics Subject Classification. 22A05 22A10 54H20.
Keywords. dynamical systems, universal minimal systems.

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Comments. This is a summary article. The results in this article will be treated fully in an article, written jointly with B. Weiss, to be published in Geometrix and Functional Analysis (GAFA).

Copyright © 2002 Charles University and Topology Atlas. Published April 2002.