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 $η$, considered as flat connection on the trivial line bundle, the differential of the Morse complex formally associated to $X$ and $η$ 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 $η$. 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 $η$. 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.