Dynamics, Laplace transform and spectral geometry

In this paper, we consider vector fields on a closed manifold whose instantons and closed trajectories can be 'counted'. Vector fields which admit Lyapunov closed one forms belong to this class. We show that under an additional hypothesis, 'the exponential growth property', the counting functions of instantons and closed trajectories have Laplace transforms which can be related to the topology and the geometry of the underlying manifold. The purpose of this paper is to introduce and explore the concept 'exponential growth property', and to describe these Laplace transforms.