Entropy and knots
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- by John Franks and R. F. Williams PDF
- Trans. Amer. Math. Soc. 291 (1985), 241-253 Request permission
Abstract:
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.References
- R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9 J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93-95. D. Bennequin, Entrelacements et équations de Pfaff, Thése de Doctorat d’Etat, Université de Paris VII, 24 novembre 1982.
- Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz’s equations, Topology 22 (1983), no. 1, 47–82. MR 682059, DOI 10.1016/0040-9383(83)90045-9 —, Knotted periodic orbits II: Fibered knots, Low Dimensional Topology, vol. 20, Contemporary Math., Amer. Math. Soc., Providence, R.I., 1983. Adrien Douady, Noeuds et structures de contact en dimension $3$ [d’aprés Daniel Bennequin], Séminaire Bourbaki, no. 604, 1983.
- 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. MR 669378
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
- A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173. MR 573822
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 291 (1985), 241-253
- MSC: Primary 58F22; Secondary 58F11, 58F15
- DOI: https://doi.org/10.1090/S0002-9947-1985-0797057-1
- MathSciNet review: 797057