An Equicontinuity Condition for Transformation Groups

1. Definitions. These definitions are essentially those given by Gottschalk and Hedlund (cf. [3]).1 Let X be a topological space, T a topological group with identity e, and ir a mapping of X X T into X with the properties: (1) ir(x, e)=x, (2) 7r(7r(rc, h), t2)=ir(x, tit2), (3) ir is continuous. The triple (X, T, ir) is called a transformation group (or dynamical system). Henceforth we shall write ir(x, t) simply as xt; and if A C.T then xA = {xt\ tÇ£A }. The orbit of x is the set xT; the orbit closure of x, the set Cl (xT). The set A is said to be minimal under T or simply minimal, provided A is an orbit closure and A does not properly contain an orbit closure. In what follows we shall be dealing with uniform spaces; for the properties of such spaces we refer to [4]. We alter the notation in writing xa instead of Va(x) lor "the neighborhood of x of index a." The group T is called equicontinuous at *6A, provided the collection of mappings {tt'I^jT, where ir'(x)=xt} is equicontinuous at x, i.e. for each index a of A" there exists an index ß of X such that xßtQxta tor all /6T". The group T is called equicontinuous provided it is equicontinuous at each point of X. The group T is called uniformly equicontinuous provided the collection of mappings {ir,|i6^'} is uniformly equicontinuous, i.e. for each index a of A there exists an index ß of X such that xßt Cxta tor all/ 6 T and all x 6 X. Let T be a topological group and let AC.T, then A is said to be left (right) syndetic in T provided that T = AK (T = KA) for some compact subset K of T. If T is abelian these two notions coincide, and we simply say that A is syndetic. The point xÇ,X is said to be almost periodic under T provided that for each index a of X, there exists a left syndetic subset A of T such that xAQxa. A point xGJ is said to be discretely almost periodic under T provided that for each index a