Topology of Foliations given by the real part of holomorphic 1-forms

Abstract

Topology of Foliations of the Riemann Surfaces given by the real part of generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB) instead of using just one closed transversal curve as in the classical approach of the ergodic theory. In some cases the TCB 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 provided by the Abel Map, plays a key role in this work. The behavior of our system in the Fundamental Domain of that covering can be easily described in the sphere with $g$ holes. It leads to the Plane Diagram of our system. The complete combinatorial model of the flow is constructed. It is based on the Plane Diagram and g straight line flows in the planes corresponding to the $g$ canonically adjoint pairs of cycles in the Transversal Canonical Basis. These pairs do not cross each other. Making cuts along them, we come to the maximal abelian fundamental domain (associated with Abel Map and Theta-functions) instead of the standard $4g$-gon in the Hyperbolic Plane and its beautiful "flat" analogs which people used for the study of geodesics of the flat metrics with singularities. Topological splitting of the flow into torical pieces is constructed. Several mistakes are corrected. Algebraic description of transversal canonical bases on the torus with obstacles is given. The appendix is extended: a reference to a work of G. Levitt is added which allowed to prove that every foliation of this class admits a TCB.