On recurrent motions of dynamical systems in a semi-metric

S. M. Dzyuba

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 144, pp. 371-382

Résumé

\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.

Détail

Keywords: dynamical systems, semi-metric space, recurrent motions, topological manifold
Mots-clés : interrelation of motions