Asymptotic signature of a three-dimensional vector field

Consider a volume preserving vector field defined in some compact domain of 3-space and tangent to its boundary. A long piece of orbit can be made int o a knot by connecting its endpoints by some arc whose length is less than the diameter of the domain. In this paper, we study the behaviour of the si gnatures of these knots as the lengths of the pieces of orbits go to infini ty. We relate this "asymptotic signature" to the "asymptotic Hopf invariant " that has been studied by Arnold.