Simple and weakly almost periodic transformation groups

Abstract

1. Introduction. In ?2 of this paper we study the invariant measures on a compact transformation group, (X, T). We assume that if Y is a closed invariant nonempty subset of X then ( Y, T) possesses an invariant measure (complete measurability assumption). Our principal result then is that, under this assumption, C(X) decomposes into a direct sum of invariant functions and functions which have integral 0 for all invariant measures (we then call (X, T) simple) if and only if there exists an upper semicontinuous decomposition 1' of X into closed invariant sets such that (i) each M' E D' contains a unique minimal set M and (ii) (M', T) has a unique invariant measure, m. Moreover, in this case m is ergodic and m(M) 1. Some of the theorems of this section generalize results of Oxtoby [12] and of Auslander [1]. In ?3 we study weakly almost periodic (w.a.p.) transformation groups. They are defined to be those transformation groups (X, T) such that iffe C(X) then the set of T-translates off have a compact closure in the weak topology of C(X). In studying w.a.p. transformation groups, we make extensive use of the enveloping semigroups, E(X, T), of Ellis [8]. Our principal result here is that if (X, T) is