Laplace transform, dynamics and spectral geometry

Abstract

We consider vector fields $X$ on a closed manifold $M$ with rest points of Morse type. For such vector fields we define the property of exponential growth. A cohomology class $ξ\in H^1(M;\mathbb R)$ which is Lyapunov for $X$ defines counting functions for isolated instantons and closed trajectories. If $X$ has exponential growth property we show, under a mild hypothesis generically satisfied, how these counting functions can be recovered from the spectral geometry associated to $(M,g,ω)$ where $g$ is a Riemannian metric and $ω$ is a closed one form representing $ξ$. This is done with the help of Dirichlet series and their Laplace transform.