Vanishing twist near focus–focus points

We show that near a simple focus–focus value in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy–momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy–momentum map that is transversal to the line of constant energy. In contrast to this, we also show that the frequency map is non-degenerate for every point in a neighbourhood of a simple focus–focus point.