Linking numbers of measured foliations

Abstract

We generalize the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension-two foliation endowed with an invariant transverse measure. We prove that the average asymptotic linking number is given by an integral of Hopf type. Considering appropriate vector fields and measured foliations, we obtain an ergodic interpretation of the Godbillon–Vey invariant of a family of codimension-one foliations discussed by Kotschick (Symplectic and Contact Topology: Interactions and Perspectives. Eds. Y. Eliashberg, B. Khesin and F. Lalonde. American Mathematical Society, Providence, RI).