Zeta functions for gradients of closed 1-forms

Given a cohomology class ξ∈H1(M;R) on the closed connected smooth manifold M we look at vector fields v which are gradient-like with respect to ξ, i.e., they admit a Lyapunov form ω, a closed 1-form representing ξ which evaluates the vector field positively whenever v≠0. Assuming that the set of zeros of v is not too complicated and v does not admit homoclinic cycles, we define a zeta function of v, an algebraic object carrying information about the closed orbit structure of v. We show that this zeta function depends continuously on v in a reasonable sense and discuss relations to chain homotopy equivalences between Novikov complexes.