Recurrence theorems for dynamical systems in a sequentially compact topological space with invariant Lebesgue measure

A property is presented that characterizes quite fully the interrelation of motions of a dynamical system $g^t$ defined in a Hausdorff sequentially compact topological space $\Gamma.$ It is noted that in the space $\Gamma$ with an invariant (with respect to $g^t$) Lebesgue measure $\mu,$ a direct analogue of the Poincare--Caratheodory recurrence theorem for sets is valid. In addition, it is shown that if $\bar{\mathcal{M}}$ is the closure of the union $\mathcal{M}$ of all minimal sets of the space $\Gamma,$ then $\mu\bar{\mathcal{M}}=\mu\Gamma,$ and through each point $p\notin\mathcal{M}$ there passes a motion $f(t,p)$ that is both positively and negatively asymptotic with respect to the compact minimal sets $\Omega_p\subset\mathcal{M}$ and $\mathrm{A}_p\subset\mathcal{M}.$ If $\Gamma$ satisfies the second axiom of countability, then $\mu\mathcal{M}=\mu\Gamma,$ i.~e. in $\Gamma,$ there is an important addition to the Poincare-Caratheodory theorem on the points recurrence.