Novikov-Morse theory for dynamical systems

Huijun Fan and Jürgen Jost

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Submission date: 14. Nov. 2000
Pages: 65
published in: Calculus of variations and partial differential equations, 17 (2003) 1, p. 29-73 
DOI number (of the published article): 10.1007/s00526-002-0159-8
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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 , (generalized) -flow for short, where 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 -Morse-Smale flows that allow the existence of "cycles" in contrast to the usual Morse-Smale flows. -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 , we construct a so-called -Morse decomposition for an -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 is a coboundary) as well as the Novikov type inequalities ( when is a nontrivial cocycle).

[8] R. Churchill, Invariant sets which carry cohomology. J. Diff. Equ. 13(1973), 523-550
[13] M. Farber, Conting zeros of closed 1-forms, Math. DG/9903133 23Mar1999