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 infinitedimensional 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 determine the asymptotic slice genus and relate it

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... the asymptotic signature. Mathematics Subject Classification (2010). 57M27, 37A05. Keywords. Asymptotic link invariant, concordance invariant, ergodic vector field. Lemma 2. Let a, n, m be natural numbers, 1 Ä a Ä n; m. There exists a natural number b, such that the link z K.n; m/ can be transformed into the link z K.a; b/ by a sequence of z Ä m C n C mn a C am saddle point moves. Proof. There exist (unique) natural numbers k, r with r < a and n D ak C r. In the following, the symbol K x ! L means that the link K can be transformed into the link L by a sequence of x saddle point moves, at most. The proof of Lemma 1 implies z K.n; m/ m ! z K.ak; m/ t z K.r; m/ y ! z K.a; km/ t z K.r; m/; Vol. 86 (2011) Asymptotic concordance invariants for ergodic vector fields 11 where y D .k 1/m C .k 1/a. By smoothing all the crossings of z K.r; m/, we have z K.a; km/ t z K.r; m/ rm ! z K.a; km/: Altogether these arrows imply z K.n; m/ z ! z K.a; b/;