  We analyse the asymptotical growth of Vassiliev invariants on non-periodic
flow lines of ergodic vector fields on domains of $\R^3$. More precisely, we
show that the asymptotics of Vassiliev invariants is completely determined by
the helicity of the vector field. As an application, we determine the
asymptotic Alexander and Jones polynomials and give a formula for the
asymptotic Kontsevich integral.
