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Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center
 

Summary: Topological Methods in Nonlinear Analysis
Journal of the Juliusz Schauder Center
Volume 00, 0000, 1-8
ASYMPTOTICALLY STABLE ONE-DIMENSIONAL COMPACT
MINIMAL SETS
Konstantin Athanassopoulos
Abstract. It is proved that an asymptotically stable, 1-dimensional, 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, 1-dimensional, 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 PoincarŽe-Bendixson theory of 1-dimensional 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 1-dimensional 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

  

Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete

 

Collections: Mathematics