Smale flows on the three-sphere
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- Trans. Amer. Math. Soc. 303 (1987), 283-310 Request permission
Abstract:
In this paper, a complete classification of Smale flows on ${S^3}$ is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstract Lyapunov graph $L’$ satisfying this set of properties, it is possible to realize a Smale flow on ${S^3}$ that has a graph $L$ as its Lyapunov graph where $L$ is equal to $L’$ up to topological equivalence. The techniques employed in proving that the conditions imposed on the graph are necessary involve some use of homology theory. Geometrical methods are used to construct the flow on ${S^3}$ associated to the given graph and therefore establish the sufficiency of the above conditions. The main theorem in this paper generalizes a result of Franks [8] who classified nonsingular Smale flows on ${S^3}$.References
- Daniel Asimov, Round handles and non-singular Morse-Smale flows, Ann. of Math. (2) 102 (1975), no. 1, 41–54. MR 380883, DOI 10.2307/1970972
- Béla Bollobás, Graph theory, Graduate Texts in Mathematics, vol. 63, Springer-Verlag, New York-Berlin, 1979. An introductory course. MR 536131
- Rufus Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations 12 (1972), 173–179. MR 336762, DOI 10.1016/0022-0396(72)90012-5
- Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. MR 458492, DOI 10.2307/1971159
- Joe Christy, Branched surfaces and attractors. I. Dynamic branched surfaces, Trans. Amer. Math. Soc. 336 (1993), no. 2, 759–784. MR 1148043, DOI 10.1090/S0002-9947-1993-1148043-8
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
- 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 —, Nonsingular Smale flows on ${S^3}$, Topology 24 (1985).
- Charles C. Pugh and Michael Shub, Suspending subshifts, Contributions to analysis and geometry (Baltimore, Md., 1980) Johns Hopkins Univ. Press, Baltimore, Md., 1981, pp. 265–275. MR 648471
- Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
- Stephen Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199–206. MR 133139, DOI 10.2307/1970311
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- R. F. Williams, Classification of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120–153; errata, ibid. (2) 99 (1974), 380–381. MR 331436, DOI 10.2307/1970908
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 303 (1987), 283-310
- MSC: Primary 58F15
- DOI: https://doi.org/10.1090/S0002-9947-1987-0896023-7
- MathSciNet review: 896023