On the interrelation of motions of dynamical systems in separable locally compact metric space with invariant measure

In this paper, we study the interrelation between recurrent and outgoing motions of dynamical systems. An outgoing motion is a motion whose $\alpha$- and $\omega$-limit sets are either empty or non-compact. It is shown that in a separable locally compact metric space $\Sigma$ with invariant Carathéodory measure, almost all points lie on trajectories of motions that are either recurrent or outgoing, i. e. in the space $\Sigma,$ the set of points $\Gamma$ lying on the trajectories of non-outgoing and non-recurrent motions has measure zero. Moreover, any motion located in $\Gamma$ is both positively and negatively asymptotic with respect to the corresponding compact minimal sets. The proof of this assertion essentially relies on the classical Poincaré–Carathéodory and Hopf recurrence theorems. From this proof and Hopf's theorem, it follows that in a separable locally compact metric space, there can exist non-recurrent Poisson-stable motions, but all these motions must necessarily be outgoing. At the same time, in the compact space $\Sigma$ any Poisson-stable motion is recurrent.