Universal Minimal Sets1
نویسنده
چکیده
Let (X, T) and (F, T) be transformation groups with the same phase group T. A homomorphism f: X—» F is a continuous mapping such that (xt)f=(xf)t (ïGZ, tÇUT). A transformation group (X, T) is called a minimal set if (xT)~ = cls[xt/tÇzT]=X(xÇzX). In this note it will be proved that given any abstract group T there exists a minimal set (M, T) with compact phase space M such that any minimal set (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 x(£M, tÇiT with t^e then xt^x. For a more complete discussion of several notions involved above see [2] and [3]. Definition 1. The ß-compactification as a transformation group. Let T be a discrete group, let ßT be the ß-compactification of T, and let t<E.T. Then the map j—»si of T into ßT is continuous and so may be extended to a map of ßT into ßT. Thus each element of T may be identified with a homeomorphism of ßT onto ßT. Under this identification (ßT, 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.
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