Asymtotically stable one-dimensional compact minimal sets

Konstantin Athanassopoulos

DOI: http://dx.doi.org/10.12775/TMNA.2007.035

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.

Keywords


Continuous flow; asymptotically stable; minimal set; isolated invariant set

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