Lyapunov exponents from unstable periodic orbits.

We propose a method that allows us to analytically compute the largest Lyapunov exponent of a Hamiltonian chaotic system from the knowledge of a few unstable periodic orbits (UPOs). In the framework of a recently developed theory for Hamiltonian chaos, by computing the time averages of the metric tensor curvature and of its fluctuations along analytically known UPOs, we have re-derived the analytic value of the largest Lyapunov exponent for the Fermi-Pasta-Ulam-beta (FPU-beta) model. The agreement between our results and the Lyapunov exponents obtained by means of standard numerical simulations confirms the point of view which attributes to UPOs the special role of efficient probes of general dynamical properties, among them chaotic instability.