 
Summary: Topological Methods in Nonlinear Analysis
Journal of the Juliusz Schauder Center
Volume 00, 0000, 18
ASYMPTOTICALLY STABLE ONEDIMENSIONAL COMPACT
MINIMAL SETS
Konstantin Athanassopoulos
Abstract. It is proved that an asymptotically stable, 1dimensional, compact
minimal set A of a continuous flow on a locally compact, metric space X is a periodic
orbit, if X is locally connected at every point of A. So, if the intrinsic topology of
the region of attraction of an isolated, 1dimensional, compact minimal set A of
a continuous flow on a locally compact, metric space is locally connected at every
point of A, then A is a periodic orbit.
1. Introduction
This note is concerned with PoincarŽeBendixson theory of 1dimensional compact
minimal sets in general locally compact, metric spaces. We are motivated by the question
how the qualitative bahaviour of a continuous flow near a compact minimal set affects its
structure. More precisely, we are interested in finding conditions refering to the flow near
a 1dimensional compact minimal set, which imply that it is a periodic orbit. Results
in this direction, have been proved in [1] for almost periodic minimal sets. The almost
periodicity is a rather restrictive internal property, which is equivalent to saying that the
