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

• Autorzy: E. Glasner , B. Weiss
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 37B10: Symbolic dynamics
• 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2003
• Tom: 176
• Numer: 3
• Strony: 277-289

Daty

wydano

2003

Twórcy

autor

E. Glasner

• Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel

autor

B. Weiss

• Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel

Identyfikatory

DOI

10.4064/fm176-3-6