On recurrent motions of dynamical systems in a semi-metric space

The present paper is devoted to studying the properties of  recur\-rent
motions of a dynamical system $g^t$ defined in a Hausdorff semi-metric space
$\Gm.$

\noindent Based on the definitions of a minimal set and recurrent motion introduced by G.
Birkhoff at the beginning of the last century, a new sufficient condition for
the recurrence of motions of the system $g^t$ in $\Gm$ is obtained. This
condition establishes a new property of motions, which rigidly connects
arbitrary and recurrent motions. Based on this property, it is shown that
if in the space $\Gm$ positively (negatively) semi-trajectory of some motion is
relative\-ly sequentially compact, then the $\om$-limit ($\al$-limit) set of
this motion is a sequentially compact minimal set.

\noindent As one of the applications of the results obtained, the behavior of motions
of the dynamical system $g^t$ given on a topological manifold $V$ is studied. This
study made it possible to significantly simplify the classical concept of
interrelation of motions on $V$ which was actually stated by G. Birkhoff in
1922 and has not changed since then.