The ω-limit sets of alternating systems

Abstract

Let f and g be elements of with . We study ω-limit sets generated by alternating trajectories of the form . In particular, we prove the following:1. A non-empty subset E of I is an ω-limit set for a continuous alternating system defined on I if and only if E is either a closed nowhere dense set or a union of finitely many disjoint non-degenerate closed intervals.2. The space of ω-limit sets is closed with respect to the Hausdorff metric.