О взаимоотношении движений динамических систем в сепарабельном локально компактном метрическом пространствес инвариантной мерой

Abstract

In this paper, we study the interrelation between recurrent and outgoing motions of dynamical systems. An outgoing motion is a motion whose α- and ω-limit sets are either empty or non-compact. It is shown that in a separable locally compact metric space Σ with invariant Caratheodory measure, almost all points lie on trajectories of motions that are either recurrent or outgoing, i. e. in the space Σ, the set of points Γ lying on the trajectories of nonoutgoing and non-recurrent motions has measure zero. Moreover, any motion located in Γ is both positively and negatively asymptotic with respect to the corresponding compact minimal sets. The proof of this assertion essentially relies on the classical Poincare-Caratheodory 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 Σ any Poisson-stable motion is recurrent.