Asymptoticity in topological dynamics

Abstract

This paper treats the notion of asymptoticity in topological transformation groups. By a topological transformation group we shall mean a topological group (the phase group) of homeomorphisms of a topological space (the phase space) onto itself, which contains the identity transformation as group identity and which has the property that any two homeomorphisms applied successively yield the same result as their group product applied once. The notion of asymptoticity has been discussed in the past only in the setting where the phase group was either the integers or the one-parameter group of reals (dynamical systems). In such systems two distinct points are said to be positively asymptotic provided their orbits under the set of positive integers (or reals), i.e., under a so-called positive ray, as homeomorphisms become and remain as close as one chooses. A similar notion is available for negative asymptoticity, i.e., asymptoticity under a negative ray; and if a pair of points is both positively and negatively asymptotic, they are said simply to be asymptotic. It is the purpose of this paper to extend the notion of asymptoticity to transformation groups with a more general structure. In ?1 we give preliminary definitions, which in the form here presented are due to Gottschalk and Hedlund [3](2). In ?2 we define an extended notion of asymptoticity relative to a replete semigroup, i.e., a semigroup of the phase group which contains a translate of each compact set in the phase group. A generalization of aand co-limit sets is available to us from [3, sec. 6.33] and is called the P-limit set of a point, where P refers to a replete semigroup. We restrict our attention largely to separable generative phase groups and prove a number of theorems about P-asymptoticity. It is shown that the P-limit set of a point has a minimality property in that it is the smallest set to which the point can be P-asymptotic, and further that the point actually is P-asymptotic to its P-limit set if and only if it does not belong to its P-limit set. The principal result of this section is a theorem about unstable transformation groups which states that under suitable hypotheses the existence of P-asymptotic points is guaranteed. This theorem contains as a special case a theorem due to Utz [6]. In ?3 we consider a symbolic transformation group, which is an extension