Novikov-Morse theory for dynamical systems

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 alpha, (generalized) alpha-flow for short, where alpha is a continuous cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define alpha-Morse-Smale flows that allow the existence of "cycles" in contrast to the usual Morse-Smale flows. alpha-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 alpha, we construct a so-called pi-Morse decomposition for an alpha-flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites. extending those of M. Farber [12]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when alpha is a coboundary) as well as the Novikov type inequalities (when alpha is a nontrivial cocycle).