Novikov-Morse theory for dynamical systems

摘要: The present paper contains an interpretation and generalization of Novikov's theory for Morse type inequalities closed 1-forms in terms concepts from Conley's dynamical systems. We introduce the concept a flow carrying cocycle \(\alpha\), (generalized) \(\alpha\)-flow short, where \(\alpha\) is continuous bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as trivial cocycle. also define \(\alpha\)-Morse-Smale that allow existence “cycles” contrast to usual Morse-Smale flows. \(\alpha\)-flows without fixed points carry not only cocycle, but class, sense [8], we shall deduce vanishing theorem generalized Novikov numbers situation. By passing suitable cover underlying compact polyhedron adapted construct so-called \(\pi\)-Morse decomposition \(\alpha\)-flow. On this basis, use Conley index derive Novikov-Morse inequalitites, extending those M. Farber [12]. In particular, these include both classical (corresponding case when coboundary) well ( nontrivial cocycle).