### Asymtotically stable one-dimensional compact minimal sets

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.

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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