Heteroclinic Curves of Gradient-like Diffeomorphsms and the Topology of Ambient Manifolds

In the study of deterministic processes described by the Morse-Smale systems noncompact heteroclinic curves play special role. These curves belong to intersections of stable and unstable manifolds of saddle periodic points. In particular, these curves are mathematical models of magnetic separators in the plasma field. We consider the class of gradient-like diffeomorphisms on three-dimensional manifolds such that their periodic points and a part of their invariant manifolds form disjoint tamely embedded surfaces. We prove that the number of the surfaces is finite and all of them have the same genus. The main result is presentation of the exact lower estimation for the number of heteroclinic curves of any diffeomorphism from considered class. This estimation is defined by genus of surfaces and their number. In addition the paper describes the topological type of manifolds which admit considered diffeomorphisms.