On the Proximal Relation in Topological Dynamics

Let iX, T) be a transformation group with compact Hausdorff phase space X. The points x and y of A" are said to be proximal provided, whenever ß is a member of the unique compatible uniformity of X, there exists i£ T such that (x/, yt) GßIf 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 xGX", let P(x) = [y(EAj (x, y)(EP]. If x£ X, the orbit of x is the set xT= [xt\ t G T], The closure of xT, denoted by ixT)~ 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 xGAf. 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 h and t2 of T if xti = xt2 for all xGA".) Let E be the closure of T in Xx. £ 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 [l]. A nonempty subset I of £ is called a right ideal in E if IE CI. 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 [l] which we shall repeatedly use in this paper.