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Templates and train tracks


Author: George Frank
Journal: Trans. Amer. Math. Soc. 308 (1988), 765-784
MSC: Primary 58F15; Secondary 57M15, 57N10, 58F25
DOI: https://doi.org/10.1090/S0002-9947-1988-0951627-9
MathSciNet review: 951627
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Abstract: Within the context of Smale flows on compact manifolds, this article deals with a relationship between abstract templates, branched $ 1$-manifolds (train tracks), and laminations representing unstable separatrices of basic sets. We show that an abstract template, the richest in information of the above three entities, determines a member of each of the remaining two groups, and partial determinations in other directions are developed. As a result of this relationship, an obstruction to the realization of certain abstract templates in nonsingular Smale flows on homology $ 3$-spheres is raised.


References [Enhancements On Off] (What's this?)

  • [1] R. Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations 12 (1972), 173-179. MR 0336762 (49:1535)
  • [2] C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R. I., 1978. MR 511133 (80c:58009)
  • [3] G. Frank, Templates, branched one-manifolds, and laminations, Ph.D. Thesis, Northwestern Univ., 1985.
  • [4] J. Franks, Symbolic dynamics inflows on three-manifolds, Trans. Amer. Math. Soc. 279 (1983), 231-236. MR 704612 (84h:58112)
  • [5] -, Flow equivalence of subshifts of finite type, Ergodic Theory and Dynamical Systems 4 (1984), 53-66. MR 758893 (86j:58078)
  • [6] -, Nonsingular Smale flows on $ {S^3}$, I.H.E.S. preprint, 1983.
  • [7] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 744-817. MR 0228014 (37:3598)
  • [8] R. Williams, Classification of one-dimensional attractors, Proc Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 341-361. MR 0266227 (42:1134)
  • [9] -, Expanding attractors, Publ. Math. Inst Hautes Études Sci. 43 (1974), 161-203.
  • [10] W. Wilson, Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413-428. MR 0251747 (40:4974)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0951627-9
Article copyright: © Copyright 1988 American Mathematical Society

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