Franks, John; Williams, R. F. Entropy and knots. (English) Zbl 0587.58038 Trans. Am. Math. Soc. 291, 241-253 (1985). We show that a smooth flow on \(S^ 3\) with positive topological entropy must possess periodic closed orbits in infinitely many different knot type equivalence classes. Cited in 19 Documents MSC: 37G99 Local and nonlocal bifurcation theory for dynamical systems 57M25 Knots and links in the \(3\)-sphere (MSC2010) 28D20 Entropy and other invariants 37C10 Dynamics induced by flows and semiflows 37A99 Ergodic theory Keywords:smooth flow on \(S^ 3\); periodic closed orbits PDFBibTeX XMLCite \textit{J. Franks} and \textit{R. F. Williams}, Trans. Am. Math. Soc. 291, 241--253 (1985; Zbl 0587.58038) Full Text: DOI References: [1] R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309 – 319. · Zbl 0127.13102 [2] J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93-95. [3] D. Bennequin, Entrelacements et équations de Pfaff, Thése de Doctorat d’Etat, Université de Paris VII, 24 novembre 1982. [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] -, Knotted periodic orbits II: Fibered knots, Low Dimensional Topology, vol. 20, Contemporary Math., Amer. Math. Soc., Providence, R.I., 1983. [6] Adrien Douady, Noeuds et structures de contact en dimension \( 3\) [d’aprés Daniel Bennequin], Séminaire Bourbaki, no. 604, 1983. · Zbl 0522.53034 [7] John M. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R. I., 1982. · Zbl 0497.58018 [8] John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. · Zbl 0515.34001 [9] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137 – 173. · Zbl 0445.58015 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.