Novikov-Morse theory for dynamical systems

Abstract

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).