Topology of the Generic Hamiltonian Foliations on the Riemann Surface

Abstract

Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces given by the real part of the generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB). 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 the 2-tori ''with obstacles''. The Fundamental Semigroup of positive closed paths transversal to foliation, and the topological characteristics of trajectories, are studied. This work contains an improved exposition of the results presented in the authors recent preprint (arXiv math.GT/0105338) and new results calculating all TCB in the 2-torus with obstacle, in terms of Continued Fractions.