  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.
