Non-singular Morse–Smale flows on n-manifolds with attractor–repeller dynamics

In the present paper the exhaustive classification up to topological equivalence of non-singular Morse–Smale flows on n -manifolds M n with exactly two periodic orbits is presented. Denote by G 2 ( M n ) the set of such flows. Let a flow f t : M n → M n belongs to the set G 2 ( M n ). Hyperbolicity of periodic orbits of f t implies that among them one is an attracting and the other is a repelling orbit. Due to the Poincaré–Hopf theorem, the Euler characteristic of the ambient manifold M n is zero. Only the torus and the Klein bottle can be ambient manifolds for f t in case of n = 2. The authors established that there are exactly two classes of topological equivalence of flows in G 2 ( M 2 ) if M 2 is the torus and three classes if M 2 is the Klein bottle. For all odd-dimensional manifolds the Euler characteristic is zero. However, it is known that an orientable three-manifold admits a flow from G 2 ( M 3 ) if and only if M 3 is a lens space L p , q . In this paper it is proved that every set G 2 ( L p , q ) contains exactly two classes of topological equivalence of flows, except the case when L p , q is homeomorphic to the three-sphere S 3 or the projective space R P 3 , where such a class is unique. Also, it is shown that the only non-orientable n -manifold (for n > 2), which admits flows from G 2 ( M n ) is the twisted I-bundle over the ( n − 1)-sphere S n − 1 × ~ S 1 . Moreover, there are exactly two classes of topological equivalence of flows in G 2 ( S n − 1 × ~ S 1 ) . Among orientable n -manifolds only the product of the ( n − 1)-sphere and the circle S n − 1 × S 1 can be ambient manifolds for flows from G 2 ( M n ) and G 2 ( S n − 1 × S 1 ) splits into two topological equivalence classes.