Almost Periodic Points with Respect to Transformation Semi-Groups

Abstract

Let X denote a topological space, let T denote a topological space in which a multiplicative binary operation is defined, and let f be a transformation of X X T into X. We agree to write ft(x) in place of f(x, t), (x e X, t e T), whenever we wish. Furthermore, let f define a transformation semi-group, that is to say, suppose f'(ft(x)) = ft8(x), (x e X; t, s e T). A point x of X is said to be almost periodic provided that if U is a neighborhood of x, then there exists a compact set A in T such that each left translate of A contains an element t for which f(x, t) e U. Strengthening the topology in T strengthens the notion of almost periodic point, the strongest type of almost periodicity occurring when T is provided with the discrete topology. We shall assume throughout that X is regular and that f is continuous on X X T. The set f(x, T) is called the orbit of the point x and the set f(x, T), denoted by F(x), is called the orbit-closure of x. Clearly, y e r(x) implies r(y) C r(x); and if x is almost periodic, then x e1 r(x). THEOREM 1. In order that x be almost periodic it is necessary that r(x) be a minimal orbit-closure; in case r(x) is compact and contains x, this condition is also sufficient. PROOF. Suppose x is almost periodic and 1r(x) is not minimal. For some point p of r(x), we have r(x) r 1(y) C r(x), whence x e r(y). Choose a neighborhood U of x so that U f r(y) = 9. There exists a compact set A in T such that each left translate of A contains an element t for which f(x, t) E U. Making use of the continuity of f and the compactness of A, we can select a neighborhood V of y so that f(V, A) C X U. For some element s of T, f(x, s) e V and for some element r of A, f(x, sr) E U. This contradicts f(x, sr) E f(V, r) C X-U. The necessity is proved. Suppose r(x) contains x and is a compact minimal orbit-closure. Let U be a neighborhood of x. For each point y of r(x) we can find a neighborhood V2, of y and an element t, in T so that f(V,, ,t) C U. There exist finitely many points yi, *, y,, of r(x) such that r(x) C U=,1 V, . Suppose t e T. Since f(x, t) E Va,, for some i, f(x, ttj,) e U. The sufficiency is proved. COROLLARY 1. If X is compact and if x is almost periodic, then x is almost periodic with respect to the discrete topology in T. COROLLARY 2. If X is compact and if x is almost periodic, then each point of r(x) is almost periodic. We say that f is pointwise almost periodic provided that each point of X is almost periodic. COROLLARY 3. In order that f be pointwise almost periodic it is necessary that the collection of orbit-closures be a partition of X; in case X is compact, this condition is also suffiient.