

Preprint 77/2000
Novikov-Morse theory for dynamical systems
Huijun Fan and Jürgen Jost
Contact the author: Please use for correspondence this email.
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
Bibtex
Download full preprint: PDF (649 kB), PS ziped (307 kB)
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