Topology of generic Hamiltonian foliations on Riemann surfaces

The topology of generic Hamiltonian dynamical systems given by the real parts of generic holomorphic 1-forms on Riemann surfaces is studied. Our approach is based on the notion of transversal canonical basis of cycles. This approach allows us to present a convenient combinatorial model of the whole topology of the flow, especially effective for $g=2$. A maximal abelian covering over the Riemann surface is needed here. The complete combinatorial model of the flow is constructed. It consists of the plane diagram and $g$ straight-line flows in 2-tori “with obstacles.” The fundamental semigroup of positive closed paths transversal to the foliation is studied. This work contains an improved exposition of the results presented in the author's recent preprint and new results concerning the calculation of all transversal canonical bases of cycles in the 2-torus with obstacle in terms of continued fractions.