Characterizations of almost periodic transformation groups

Abstract

Let X be a uniform space, let T be a multiplicative topological group, and let T act as a transformation group on X. A subset A of T is said to be (left) syndetic provided that T= AK for some compact subset K of T. T he transformation group T is said to be almost periodic on X provided that if a is an index of X, then there exists a syndetic subset A of T such that xA Cxa for all xEX. If xGX, then the transformation group T is said to be locally almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x and a syndetic subset A of T such that VA CU. The transformation group T is said to be locally almost periodic on X in case T is locally almost periodic at every point of X. If xEX, then the transformation group T is said to be locally weakly almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x, a syndetic subset A of T, and a compact subset C of T such that y E V implies the existence of a subset B of T for which A CBC and yBC U. It is readily proved that if xcX, then T is locally weakly almost periodic at x if and only if for each neighborhood U of x there exist a neighborhood V of x and a compact subset K of T such that VTC UK. If xEX, then the transformation group T is said to be equicontinuous at x provided that if a is an index of X, then there exists an index e of X such that xftCxta for all tC T. The transformation group T is said to be equicontinuous on X in case T is equicontinuous at every point of X. The transformation group T is said to be uniformly equicontinuous on X provided that if a is an index of X, then there exists an index ,B of X such that xl3tCxta for all xEX and all tcT. It is readily verified that if X is compact, then T is uniformly equicontinuous if and only if T is equicontinuous. The transformation group T is said to be distal on X provided that if x, yEX with x5-zy, then there exists an index a of X such that (xt, yt) Ea for all t E T. We also consider T to be a transformation group acting on X X X in the following manner: if x, yEX and if tET, then (x, y)t is defined to be (xt, yt).