Non-singular Morse–Smale flows on 3-dimensional manifolds

The classical Lefschetz formula expresses a relationship between the fixed points of certain transformations and homological invariants. When applied to continuous systems, one obtains the Hopf formula relating the singularities of a flow to the Euler characteristic of the underlying manifold. The main goal of this paper is to prove an analogous formula that relates the closed orbits of a stable flow to a homological invariant of the manifold. There are serious obstruction to finding such a formula for flows. The first is that there are flows on any closed manifold of dimension > 4 of zero Euler characteristic with no periodic orbits whatsoever [27]. These flows, however, are unstable by Pugh's C a closing lemma [19]. So it is natural to restrict attention to flows that satisfy Axiom A and the No Cycles property, since Smale has shown that these flows are O-stable [23]. Secondly there are gradient flows on any closed manifold that are Axiom A-No Cycles and have no closed orbits [24]. So it is natural to study only nonsingular flows. The possibility that a Lefschetz type formula for such flows existed was suggested to us by R. Bowen. Even nonsingular Morse-Smale flows, however, are difficult to deal with since Asimov's geometric index is not an invariant of the nonsingular homotopy class [2]. We must further restrict the class of flows to assume that the closed orbits relate in some manner to the homology of the underlying manifold. We define in Sect. 2 the class of homology proper flows. Such a flow has the property that closed orbits of large period lie in large homology classes. In Sect. 7 we show that a homology proper Axiom A-No Cycles flow does satisfy a Lefschetz-type formula that relates invariants of the closed orbits to a manifold invariant. This formula is closely related to Franks' study of Smale flows on S 3 in which the linking numbers of closed orbits are used to compute the Alexander polynomial of a certain link. Indeed the manifold invariant used in Sect. 7 is a close analogue of an Alexander polynomial that we call the Alexander

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott–Ruelleresonant states associated with Anosov and Morse–Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse–Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.

Any link in a 3-manifold is the closed orbits of a non-singular Morse-Smale flow after taking the split sum with the unlink and the connected sum with S2 × S1s.

The topological equivalence of low-dimensional Morse-Smale flows without fixed point (NMS-flows) under assumptions of various generality is the subject of a number of publications. Starting from dimension 4, there are only few results on classification. However, it is known that there exists nonsingular flows with wildly embedded invariant saddle manifolds. In this paper the class of nonsingular Morse-Smale flows on closed orientable 4-manifolds with a unique saddle orbit which is, moreover, nontwisted, is considered. It is shown that the equivalence class of a certain knot embedded in $\mathbb S^2\times\mathbb S^1$ is a complete invariant of such a flow. Given a knot in $\mathbb S^2\times\mathbb S^1$, a standard representative in the class of flows under consideration is constructed. The supporting manifold of all such flows is shown to be the manifold $\mathbb S^3\times\mathbb S^1$. Bibliography: 24 titles.

The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.

We consider the optimal flows on noncompact surfaces with boundary, which have a minimum number of fixed points and all these points lie on the boundary of the surface. It is proved that the flow is optimal if it has a single sink and a single source. We describe the structures of the optimal flows on a simply connected region, on a Mobius strip, on a torus with hole, and on a Klein bottle with hole.