Fractional Hamiltonian Monodromy from a Gauss-manin Monodromy

Fractional Hamiltonian Monodromy is a generalization of the notion of Hamiltonian Monodromy, recently introduced by N. N. Nekhoroshev, D. A. Sadovskíı and B. I. Zhilinskíı for energy-momentum maps whose image has a particular type of non-isolated singularities. In this paper, we analyze the notion of Fractional Hamiltonian Monodromy in terms of the Gauss-Manin Monodromy of a Riemann surface constructed from the energy-momentum map and associated to a loop in complex space which bypasses the line of singularities. We also prove some propositions on Fractional Hamiltonian Monodromy for 1 : −n and m : −n resonant systems.