Dynamics near the p : − q Resonance

We study the dynamics near the truncated p : ±q resonant Hamiltonian equilibrium for p, q coprime. The critical values of the momentum map of the Liouville integrable system are found. The three basic objects reduced period, rotation number, and non-trivial action for the leading order dynamics are computed in terms of complete hyperelliptic integrals. A relation between the three functions that can be interpreted as a decomposition of the rotation number into geometric and dynamic phase is found. Using this relation we show that the p : −q resonance has fractional monodromy. Finally we prove that near the origin of the 1 : −q resonance the twist vanishes.