The universal minimal system for the group of homeomorphisms of the Cantor set

E. Glasner ; B. Weiss

Fundamenta Mathematicae, Tome 177 (2003), p. 277-289 / Harvested from The Polish Digital Mathematics Library

Résumé

Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a 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. We show that for the topological group G = Homeo(E) of self-homeomorphisms of the Cantor set E, with the topology of uniform convergence, the universal minimal system (M(G),G) is isomorphic to Uspenskij’s “maximal chains” dynamical system (Φ,G) in $2^{2^E}$. In particular it follows that M(G) is homeomorphic to the Cantor set. Our main tool is the “dual Ramsey theorem”, a corollary of Graham and Rothschild’s Ramsey’s theorem for n-parameter sets. This theorem is used to show that every minimal symbolic G-system is a factor of (Φ,G), and then a general procedure for analyzing G-actions of zero-dimensional topological groups is applied to show that (M(G),G) is isomorphic to (Φ,G).

Détail

Publié le : 2003-01-01

Zbl 1022.37009

EUDML-ID : urn:eudml:doc:282616