Novikov-Morse theory for dynamical systems

H. Fan; J. Jost
May 2003
Calculus of Variations & Partial Differential Equations;May2003, Vol. 17 Issue 1, p29
Academic Journal
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 FORMULA>, (generalized) FORMULA>-flow for short, where FORMULA> 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 FORMULA>-Morse-Smale flows that allow the existence of "cycles" in contrast to the usual Morse-Smale flows. FORMULA>-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 FORMULA>, we construct a so-called $\pi$-Morse decomposition for an FORMULA>-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 FORMULA> is a coboundary) as well as the Novikov type inequalities ( when FORMULA> is a nontrivial cocycle).


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