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

\noindent Abstract. The present paper is devoted to studying the properties of recurrent motions of a dynamical system $g^t$ defined in a Hausdorff semi-metric space $\Gamma.$
\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 $\Gamma$ 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 $\Gamma$ positively (negatively) semi-trajectory of some motion is relatively sequentially compact, then the $\omega$-limit ($\alpha$-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.