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.