This paper is concerned
with a flow on a metric space, and some topological properties of the set of
orbits which are asymptotic to a given invariant subset, with particular
emphasis on the flow near an invariant Jordan curve (e.g., a periodic orbit) in
an orientable n-manifold Mn. The invesligation began with the asking of
the simple question: Can a periodic orbit J of a vector field in Rn be the
ω-limit set of precisely one orbit distinct from J? It is shown that if the
periodic orbit J is a maximal element in the class of invariant continuua
lying in a neighborhood of J, then the answer is negative and in fact the
set of orbits asymptotic to J as t →∞ has some of the same topological
properties already known from the stable manifold theorems in the case of an
elementary periodic orbit of a flow generated by a smooth ordinary differential
equation.
