Contact topology and hydrodynamics. III: Knotted orbits. (English) Zbl 0960.76020
Summary: We employ the relationship between contact structures and Beltrami fields derived in part I of this series [see the authors, Nonlinearity 13, No. 2, 441-458 (2000; Zbl 0982.76021)] to construct a steady nonsingular solution to the Euler equations on a Riemannian \(S^3\) whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on \(S^3\). Sufficient review of concepts is included to make this paper independent of the previous works in this series.
MSC:
| 76B47 | Vortex flows for incompressible inviscid fluids |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
| 37J55 | Contact systems |
| 37C27 | Periodic orbits of vector fields and flows |
| 53D10 | Contact manifolds (general theory) |
Keywords:
tight contact structures; Reeb field; Euler equations; knots; templates; S(3); contact structures; Beltrami fields; steady nonsingular solution; contact-topological controls; tight contact structureCitations:
Zbl 0982.76021References:
| [1] | B. Aebischer, M. Borer, M. Kälin, Ch. Leuenberger, and H. M. Reimann, Symplectic geometry, Progress in Mathematics, vol. 124, Birkhäuser Verlag, Basel, 1994. An introduction based on the seminar in Bern, 1992. · Zbl 0932.53002 |
| [2] | Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. · Zbl 0902.76001 |
| [3] | Daniel Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87 – 161 (French). · Zbl 0573.58022 |
| [4] | Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology 22 (1983), no. 1, 47 – 82. · Zbl 0507.58038 · doi:10.1016/0040-9383(83)90045-9 |
| [5] | Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 1 – 60. · Zbl 0526.58043 · doi:10.1090/conm/020/718132 |
| [6] | V. Colin. Recollement de variétés de contact tendues. Bull. Soc. Math. France 127:43-69, 1999. CMP 99:15 |
| [7] | T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr, and A. M. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986), 353 – 391. · Zbl 0622.76027 · doi:10.1017/S0022112086002859 |
| [8] | J. Etnyre and R. Ghrist. Contact topology and hydrodynamics I: Beltrami fields and the Seifert conjecture. Nonlinearity, 13:441-458, 2000. CMP 2000:09 · Zbl 0982.76021 |
| [9] | J. Etnyre and R. Ghrist. Stratified integrals and unknots in inviscid flows. Cont. Math., 246:99-111, 1999. CMP 2000:07 · Zbl 0989.76010 |
| [10] | Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent. Math. 98 (1989), no. 3, 623 – 637. · Zbl 0684.57012 · doi:10.1007/BF01393840 |
| [11] | Yakov Eliashberg, Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165 – 192 (English, with French summary). · Zbl 0756.53017 |
| [12] | Yakov Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 171 – 193. · Zbl 0809.53033 |
| [13] | Jean-Luc Gautero, Chaos lagrangien pour une classe d’écoulements de Beltrami, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 301 (1985), no. 15, 1095 – 1098 (French, with English summary). · Zbl 0579.76063 |
| [14] | Robert Ghrist and Philip Holmes, Knots and orbit genealogies in three-dimensional flows, Bifurcations and periodic orbits of vector fields (Montreal, PQ, 1992) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 408, Kluwer Acad. Publ., Dordrecht, 1993, pp. 185 – 239. · Zbl 0803.58047 |
| [15] | Robert W. Ghrist and Philip J. Holmes, An ODE whose solutions contain all knots and links, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), no. 5, 779 – 800. · Zbl 0878.34038 · doi:10.1142/S0218127496000448 |
| [16] | Robert W. Ghrist, Branched two-manifolds supporting all links, Topology 36 (1997), no. 2, 423 – 448. · Zbl 0869.57007 · doi:10.1016/0040-9383(96)00006-7 |
| [17] | Robert W. Ghrist, Philip J. Holmes, and Michael C. Sullivan, Knots and links in three-dimensional flows, Lecture Notes in Mathematics, vol. 1654, Springer-Verlag, Berlin, 1997. · Zbl 0869.58044 |
| [18] | John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59 – 72. R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73 – 99. · Zbl 0436.58018 |
| [19] | H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515 – 563. · Zbl 0797.58023 · doi:10.1007/BF01232679 |
| [20] | Philip Holmes, Knotted periodic orbits in suspensions of Smale’s horseshoe: period multiplying and cabled knots, Phys. D 21 (1986), no. 1, 7 – 41. · Zbl 0623.58014 · doi:10.1016/0167-2789(86)90076-X |
| [21] | P. J. Holmes, Knotted periodic orbits in suspensions of annulus maps, Proc. Roy. Soc. London Ser. A 411 (1987), no. 1841, 351 – 378. · Zbl 0642.58039 |
| [22] | Philip Holmes and R. F. Williams, Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences, Arch. Rational Mech. Anal. 90 (1985), no. 2, 115 – 194. · Zbl 0593.58027 · doi:10.1007/BF00250717 |
| [23] | H. Hofer, K. Wysocki, and E. Zehnder, Unknotted periodic orbits for Reeb flows on the three-sphere, Topol. Methods Nonlinear Anal. 7 (1996), no. 2, 219 – 244. · Zbl 0898.58018 |
| [24] | De-Bin Huang, Xiao-Hua Zhao, and Hui-Hui Dai, Invariant tori and chaotic streamlines in the ABC flow, Phys. Lett. A 237 (1998), no. 3, 136 – 140. · Zbl 0959.76015 · doi:10.1016/S0375-9601(97)00789-5 |
| [25] | A. M. Candela and A. Salvatore, Light rays joining two submanifolds in space-times, J. Geom. Phys. 22 (1997), no. 4, 281 – 297. · Zbl 0880.58005 · doi:10.1016/S0393-0440(96)00034-4 |
| [26] | H. K. Moffatt, Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. I. Fundamentals, J. Fluid Mech. 159 (1985), 359 – 378. · Zbl 0616.76121 · doi:10.1017/S0022112085003251 |
| [27] | H. Moffatt. Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology: part II. J. Fluid Mech., 166:359-378, 1986. · Zbl 0639.76111 |
| [28] | Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. · Zbl 0844.58029 |
| [29] | W. Thomson. On vortex motion. Trans. R. Soc. Edin., 25:217-260, 1869. · JFM 01.0341.02 |
| [30] | R. F. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977) Springer, Berlin, 1977, pp. 94 – 112. Lecture Notes in Math., Vol. 615. |
| [31] | R. Williams. The universal templates of Ghrist. Bull. Am. Math. Soc., 35(2):145-156, 1998. CMP 98:12 · Zbl 0902.57001 |
| [32] | Xiao Hua Zhao, Keng Huat Kwek, Ji Bin Li, and Ke Lei Huang, Chaotic and resonant streamlines in the ABC flow, SIAM J. Appl. Math. 53 (1993), no. 1, 71 – 77. · Zbl 0769.76016 · doi:10.1137/0153005 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
