Symplectic Invariants Near Hyperbolic-Hyperbolic Points

We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system. We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C.~Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.