Asymtotically stable one-dimensional compact minimal sets

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.