Novikov - Morse Theory for Dynamical Systems HuiJun

The present paper contains an interpretation and generalization of Novikov’s theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley’s theory for dynamical systems. We introduce the concept of a flow carrying a cocycle α, (generalized) α-flow for short, where α is a cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define α-Morse-Smale flows that allow the existence of “cycles” in contrast to the usual Morse-Smale flows. α-flows without fixed points carry not only a cocycle, but a cohomology class, in the sense of [8], and we shall deduce a vanishing theorem for generalized Novikov numbers in that situation. By passing to a suitable cover of the underlying compact polyhedron adapted to the cocycle α, we construct a so-called π-Morse decomposition for an α-flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites, extending those of M. Farber [13]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when α is a coboundary) as well as the Novikov type inequalities ( when α is a nontrivial cocycle).