The Universal Minimal System for the Group of Homeomorphisms of the Cantor Set

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 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. In this paper 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 22 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 Gactions of zero dimensional topological groups is used to show that (M(G), G) is isomorphic to (Φ, G).