A Characterization of Distal and Point-Distal Minimal Transformation Groups

Abstract

If G is a locally compact topological group, let $BC(G)$ denote the set of real-valued, bounded, uniformly continuous functions on G with the compact-open topology. Using the fact that the distal (weakly distal) functions are the elements of $BC(G)$ whose orbit closures are compact distal (point-distal) minimal sets, we can characterize compact distal and point-distal minimal transformation groups. Let $(X,G,\phi )$ be a right transformation group where X is compact Hausdorff and minimal under G. Then X is a compact distal (point-distal) minimal set if and only if there is a point $x \in X$ such that for any homomorphism $h:X \to BC(G),h(x)$ is a right distal (weakly distal) function.