On some properties of motions of dynamical systems on compact manifolds

S. M. Dzyuba

Vestnik rossijskih universitetov. Matematika, Tome 30 (2025) no. 149, pp. 28-40

Résumé

The article considers the motions of dynamical system $g^t$ defined on a topological compact manifold $V.$ It is shown that the set $M_1$ of non-wandering points with respect to $V$ is the set of central motions $\mathfrak{M}$, and the union of all compact minimal sets is everywhere dense in the set $\mathfrak{M}.$ It is established that for any motion $f(t,p),$ there exists a compact minimal set $\Omega\subset V$ with the following property: for all values $t_0\in\mathbb{R}$ and every neighborhood $E_{\Omega}$ of the set $\Omega,$ the probability that the arc $\{f(t,p)\colon t\in[t_0,t_1]\}$ of the motion trajectory $f(t,p)$ belongs to the set $E_{\Omega},$ tends to 1 as $t_1\to+\infty;$ a similar statement is true for the arc $\{f(t,p)\colon t\in[-t_1,t_0]\}.$ All statements of this article can be transferred without any changes to the system $g^t$ defined in a Hausdorff sequentially compact topological space.

Détail

Keywords: topological manifold, dynamical systems, set of central motions, probabilistic properties of motions