×

zbMATH — the first resource for mathematics

Signature asymptotique d’un champ de vecteurs en dimension 3. (Asymptotic signature of a vector field in dimension 3). (French. English summary) Zbl 1010.37010
Summary: 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 into 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 signatures of these knots as the lengths of the pieces of orbits go to infinity. We relate this “asymptotic signature” to the “asymptotic Hopf invariant” that has been studied by V. I. Arnol’d [Sel. Math. Soc. 5, 327-345 (1986; Zbl 0623.57016)].

MSC:
37C10 Dynamics induced by flows and semiflows
37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
57R25 Vector fields, frame fields in differential topology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] R. Abraham et J. Robbin, Transversal Mappings and Flows , Benjamin, New York, 1967. · Zbl 0171.44404
[2] V. Arnold, The asymptotic Hopf invariant and its applications, Selecta Math. Soviet. 5 (1986), 327–345. · Zbl 0623.57016
[3] V. Arnold et B. Khesin, Topological Methods in Hydrodynamics, Appl. Math. Sci. 125 , Springer, New York, 1998. · Zbl 0902.76001
[4] J. Birman et R. F. Williams, Knotted periodic orbits in dynamical systems, I: Lorenz’s equations, Topology 22 (1983), 47–82. · Zbl 0507.58038 · doi:10.1016/0040-9383(83)90045-9
[5] G. Burde et H. Zieschang, Knots , de Gruyter Stud. Math. 5 , de Gruyter, Berlin, 1985.
[6] G. Contreras et R. Iturriaga, Average linking numbers, Ergodic Theory Dynam. Systems 19 (1999), 1425–1435. · Zbl 0960.37006 · doi:10.1017/S0143385799146777
[7] J.-M. Gambaudo et É. Ghys, Enlacements asymptotiques, Topology 36 (1997), 1355–1379. · Zbl 0913.58003 · doi:10.1016/S0040-9383(97)00001-3
[8] R. Ghrist, P. Holmes, et M. Sullivan, Knots and Links in Three-dimensional Flows, Lecture Notes in Math. 1654 , Springer, Berlin, 1997. · Zbl 0869.58044 · doi:10.1007/BFb0093387
[9] M. Golubitsky et V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts Math. 14 , Springer, New York, 1973. · Zbl 0294.58004
[10] C. McA. Gordon et R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), 53–69. · Zbl 0391.57004 · doi:10.1007/BF01609479 · eudml:142569
[11] F. Hirzebruch et D. Zagier, The Atiyah-Singer Theorem and Elementary Number Theory, Math. Lecture Ser. 3 , Publish or Perish, Boston, 1974. · Zbl 0288.10001
[12] L. Kauffman, On Knots, Ann. of Math. Stud. 115 , Princeton Univ. Press, Princeton, 1987.
[13] S. Łojasiewicz, Sur la géométrie semi- et sous-analytique, Ann. Inst. Fourier (Grenoble) 43 (1993), 1575–1595. · Zbl 0803.32002 · doi:10.5802/aif.1384 · numdam:AIF_1993__43_5_1575_0 · eudml:75048
[14] R. Mañé, Introdução à teoria ergódica, Projeto Euclides 14 , Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1983. · Zbl 0581.28010
[15] J. Milnor, “Infinite cyclic coverings” dans Conference on the Topology of Manifolds (E. Lansing, Mich., 1967), Prindle, Weber, and Schmidt, Boston, 1968, 115–133. · Zbl 0179.52302
[16] K. Murasugi, Knot Theory and Its Applications, traduit de l’édition japonaise (1995) par Bohdan Kurpita, Birkhäuser, Boston, 1996. · Zbl 0864.57001
[17] D. Rolfsen, Knots and Links, Math. Lecture Ser. 7 , Publish or Perish, Houston, 1990.
[18] K. Taniyama et A. Yasuhara, On \(C\)-distance of knots, Kobe J. Math. 11 (1994), 117–127. · Zbl 0846.57007
[19] A. Verjovsky et R. Vila Freyer, The Jones-Witten invariant for flows on a \(3\)-dimensional manifold, Comm. Math. Phys. 163 (1994), 73–88. · Zbl 0808.57011 · doi:10.1007/BF02101735
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.