An equicontinuity condition for transformation groups

Abstract

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 XX T into X with properties: (1) 7r(x, e) =x, (2) ir(ir(x, ti), 12) =7r(X, 1112), (3) ir is continuous. The triple (X, T, ir) is called a transformation group (or dynamical system). Henceforth we shall write 7r(x, t) simply as xt; and if ACT then xA = {xtItCA }. The orbit of x is set xT; orbit closure of x, 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 properties of such spaces we refer to [4]. We alter notation in writing xa instead of Va(x) for the neighborhood of x of index a. The group T is called equicontinuous at xEX, provided collection of mappings {i rtI tET, where 7rt(x) =xt } is equicontinuous at x, i.e. for each index a of X there exists an index i3 of X such that xitCxta for all t (=T. The group T is called equicontinuous provided it is equicontinuous at each point of X. The group T is called uniformly equicontinuous provided collection of mappings {7rtjtE T} is uniformly equicontinuous, i.e. for each index a of X there exists an index , of X such that xi3tCxta for all t CT and all xCX. Let T be a topological group and let A CT, 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 xEX 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 xA Cxa. A point xCX is said to be discretely almost periodic under T provided that for each index a