Large time asymptotics of the wave fronts length II: surfaces with integrable Hamiltonians

In the paper [Vi-20], the author proves that the length |St| of the wave front St at time t of a wave propagating in an Euclidean disk D of radius 1, starting from a source A, admits a linear asymptotics as t → +∞: |St| ∼ (2 arcsin a)t with a = d(0, A). In the paper [Co-Vi-20], we gave a more direct proof and some improvements of that result. Here, we will explain that this result is quite general for surfaces with an integrable Hamiltonian. We discuss only the 2D case for simplicity. The main idea is to use action-angle coordinates (section 2) in order to get a nice integral expression for |St| (section 4). Integrable systems have in general singularities, therefore we need to make some genericity assumptions (section 2) and to study what happens to the action-angle coordinates (section 3) near these generic singularities. We need then to evaluate some oscillatory integrals (section 6) using an ergodic lemma (Appendix B). For the geodesic flow on closed manifolds of negative curvature, Margulis [Ma-69] proved that the asymptotics of the length is exponential. The generic behaviour is not known. Here we study the integrable case which is highly non generic. Before starting, let us give a rough version of the main theorem 5.1: Let (X, g) be a 2D-Riemannian manifold. Let H : T X → R be an integrable Hamiltonian near a given energy E. Assume that the