Dynamics, Laplace transform and Spectral geometry

Abstract

We consider a vector field $X$ on a closed manifold which admits a Lyapunov one form. We assume $X$ has Morse type zeros, satisfies the Morse--Smale transversality condition and has non-degenerate closed trajectories only. For a closed one form $\eta$, considered as flat connection on the trivial line bundle, the differential of the Morse complex formally associated to $X$ and $\eta$ is given by infinite series. We introduce the exponential growth condition and show that it guarantees that these series converge absolutely for a non-trivial set of $\eta$. Moreover the exponential growth condition guarantees that we have an integration homomorphism from the deRham complex to the Morse complex. We show that the integration induces an isomorphism in cohomology for generic $\eta$. Moreover, we define a complex valued Ray--Singer kind of torsion of the integration homomorphism, and compute it in terms of zeta functions of closed trajectories of $X$. Finally, we show that the set of vector fields satisfying the exponential growth condition is $C^0$--dense.

35page

References in corpus (4)

• Euler structures, the variety of representations and the Milnor-Turaev torsion
• On the topology and analysis of a closed one form. I (Novikov's theory revisited)
• C^0-generic properties of boundary operators in Novikov Complex
• Laplace transform, dynamics and spectral geometry

Cited by in corpus (8)

• Complex valued Ray-Singer torsion
• Persistence for Circle Valued Maps
• Topology of angle valued maps, bar codes and Jordan blocks
• Graph Representations and Topology of Real and Angle Valued Maps
• Survey on analytic and topological torsion
• Smooth structure on the moduli space of instantons of a generic vector field
• New invariants for a real valued and angle valued map (an Alternative to Morse- Novikov theory)
• Dynamics, Spectral Geometry and Topology