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

E. Glasner ¹   ; B. Weiss ²

¹ Department of Mathematics Tel Aviv University Ramat Aviv, Israel
² Institute of Mathematics Hebrew University of Jerusalem Jerusalem, Israel

Fundamenta Mathematicae, Tome 176 (2003) no. 3, pp. 277-289

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 $G=\mathop {\rm Homeo}(E)$ 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 $({\mit \Phi },G)$ 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 $({\mit \Phi },G)$, 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 $({\mit \Phi },G)$.

Détail

DOI : 10.4064/fm176-3-6

Keywords: each topological group admits unique universal minimal dynamical system locally compact noncompact group nonmetrizable system rich structure which acts effectively however there topological groups which trivial one point system extremely amenable groups topological groups which metrizable space which has explicit description topological group mathop homeo self homeomorphisms cantor set topology uniform convergence universal minimal system isomorphic uspenskijs maximal chains dynamical system mit phi particular follows homeomorphic cantor set main tool dual ramsey theorem corollary graham rothschilds ramseys theorem n parameter sets theorem every minimal symbolic g system factor mit phi general procedure analyzing g actions zero dimensional topological groups applied isomorphic mit phi