ON EXTREMELY AMENABLE GROUPS OF HOMEOMORPHISMS

A topological group C is extremely amenable if every compact C-space has a C-fixed point. Let X be compact and G subset of Homeo (X). We prove that the following are equivalent: (1) C; is extremely amenable; (2) every minimal closed C-invariant subset of Exp R is a singleton, where I? is the closure of the set of all graphs of g is an element of C; in the space Exp (X-2) (Exp stands for the space of closed subsets); (3) for each n = 1, 2, ... there is a closed C-invariant subset Y-n of (Exp X)(n) such that boolean OR Y-infinity(n=1)n contains arbitrarily fine covers of X and for every n >= 1 every minimal closed C-invariant subset of Exp Y-n is a singleton. This yields an alternative proof of Pestov's theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set, (or of the interval [0,1]) is extremely amenable.