  We study the asymptotics of a family of link invariants on the orbits of a
smooth volume-preserving ergodic vector field on a compact domain of the
3-space. These invariants, called linear saddle invariants, include many
concordance invariants and generate an infinite-dimensional vector space of
link invariants. In contrast, the vector space of asymptotic linear saddle
invariants is 1-dimensional, generated by the asymptotic signature. We also
relate the asymptotic slice genus to the asymptotic signature.
