On the proximal relation in topological dynamics

Abstract

Let (X, T) be a transformation group with compact Hausdorff phase space X. The points x and y of X are said to be proximal provided, whenever : is a member of the unique compatible uniformity of X, there exists tI T such that (xt, yt) Go. If x and y are not proximal, they are said to be distal. Let P denote the proximal relation in X. P is a reflexive, symmetric, T invariant relation, but is not in general transitive or closed. As is customary, if xCX, let P(x)= [yGXj (x, y)EP]. If xeX, the orbit of x is the set xT= [xtj tE T]. The closure of xT, denoted by (xT)is called orbit closure of x. A nonempty subset M of X is said to be a minimal orbit closure, or minimal set, if M= (xT)for all xEM. If A is a nonempty closed, T invariant subset of X, then A contains at least one minimal set [3, 2.22]. We may consider T as a subset of Xx. (We identify two elements t1 and t2 of T if xt1=xt2 for all xCX.) Let E be the closure of T in Xx. E is a compact semigroup (but not a topological semigroup); it is called the enveloping semigroup of (X, T). The enveloping semigroup of a transformation group was defined in [2]. Its algebraic properties, and their connection with the recursive properties of the transformation group are studied in [1]. A nonempty subset I of E is called a right ideal in E if IECd. If I contains no proper nonempty subsets which are also right ideals, I is called a minimal right ideal. In Lemma 1, we summarize some results from [1] which we shall repeatedly use in this paper.