Recursive properties of transformation groups in hyperspaces (original) (raw)

1975, Mathematical Systems Theory

Introduction. Let (X, T) be a transformation group with compact Hausdorff phase space X and arbitrary acting group T. There is a unique uniformity f~ of X which is compatible with the topology for X. Consider the hyperspace 2 x consisting all non-empty closed subsets of X. For each E ~ 2 x and each ~ e f~, we denote F.= = U {x=Lx ~ F.} =* = {(a, S)IA, B ~2 x, .4 = B~, ~ < A=} Then oJ is a uniformity base for the hyperspace 2 x. If ~* is the uniformity generated by o~, then (2 x, f2*) is a uniform space such that the topology generated by f~* is compact Hausdorff. With this topology, the group T acts on 2 x in a natural way so that we obtain a transformation group (2 x, T) called the hypertransformation group associated with (X, T). In this paper, we first construct the transformation group (2 x, T) and obtain some of its general properties. Then we deal with the recursive properties of (2 x, T). In particular, we are interested in regular almost periodicity and almost periodicity. The main reference for information on the hyperspace 2 x is Michael [6]. For transformation groups, we follow the notation and terminology used in Gottschalk and Hedlund [4] Standing Notation and Hypothesis. Throughout this paper, the pair (X, T) will denote a transformation group with compact Hausdorff phase space X and arbitrary acting group T, and f~ will denote the unique uniformity compatible with the topology for X. The topology for the hyperspace 2 x is always the topology generated by the uniformity f~* defined above.