LYAPUNOV INSTABILITY IN KAM STABLE HAMILTONIANS WITH TWO DEGREES OF FREEDOM

For a fixed frequency vector omega is an element of R2 a {0} obeying omega 1 omega 2 < 0, we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate analytic Hamiltonian, having a Lya-punov unstable elliptic equilibrium with frequency omega. In particular, the el-liptic fixed points thus constructed will be KAM stable, i.e., accumulated by invariant tori whose Lebesgue density tend to one in the neighborhood of the point and whose frequencies cover a set of positive measure.Similar examples for near-integrable Hamiltonians in action-angle coor-dinates, in the neighborhood of a Lagrangian invariant torus with arbitrary frequency vector, are also given in this work.