Universal minimal sets

Abstract

Let (X, T) and (Y, T) be transformation groups with the same phase group T. A homomorphism f: X-Y is a continuous mapping such that (xt)f= (xf)t (xCX, tE T). A transformation group (X, T) is called a minimal if (xT)= cls [xt/tC T] = X(x CX). In this note it will be proved that given any abstract group T there exists a minimal (M, T) with compact phase space M such that any minimal (X, T) with compact X is a homomorphic image of (M, T). Furthermore this universal minimal set is unique up to an isomorphism, and given xCM, tCT with t$e then xt$x. For a more complete discussion of several notions involved above see [21 and [3 ]. DEFINITION 1. The fl-compactification as a transformation group. Let T be a discrete group, let f,T be the ,B-compactification of T, and let tG T. Then the map s-*st of T into f,T is continuous and so may be extended to a map of fBT into f,T. Thus each element of T may be identified with a homeomorphism of ,BT onto f,T. Under this identification (fiT, T) becomes a transformation group. Henceforth all transformation groups (X, T) will be assumed to have compact phase spaces, X, and discrete phase group T.