Laplace transform, dynamics and spectral geometry

Original · EN

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∈ H¹(M;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.