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Article

О существовании периодических траекторий для непрерывных потоков Морса-Смейла

We consider the class of continuous Morse-Smale flows defined on a topological closed manifold $M^n$ of dimension n which is not less than three, and such that the stable and unstable manifolds of saddle equilibrium states do not have intersection. We establish a relationship between the existence of such flows and topology of closed trajectories and topology of ambient manifold. Namely, it is proved that if $f^t$ (that is a continuous Morse-Smale flow from considered class) has mu sink and source equilibrium states and $\nu$ saddles of codimension one, and the fundamental group $\pi_{1}(M^n$) does not contain a subgroup isomorphic to the free product $g = 1/ 2 ( \nu−\mu  + 2)$ copies of the group of integers Z , then the flow $f^t$ has at least one periodic trajectory.