Templates and train tracks
HTML articles powered by AMS MathViewer
- by George Frank PDF
- Trans. Amer. Math. Soc. 308 (1988), 765-784 Request permission
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
- 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
- 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 G. Frank, Templates, branched one-manifolds, and laminations, Ph.D. Thesis, Northwestern Univ., 1985.
- John Franks, Symbolic dynamics in flows on three-manifolds, Trans. Amer. Math. Soc. 279 (1983), no. 1, 231–236. MR 704612, DOI 10.1090/S0002-9947-1983-0704612-1
- John Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53–66. MR 758893, DOI 10.1017/S0143385700002261 —, Nonsingular Smale flows on ${S^3}$, I.H.E.S. preprint, 1983.
- 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 one dimensional attractors, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 341–361. MR 0266227 —, Expanding attractors, Publ. Math. Inst Hautes Études Sci. 43 (1974), 161-203.
- F. Wesley Wilson Jr., Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413–428. MR 251747, DOI 10.1090/S0002-9947-1969-0251747-9
Additional Information
- © Copyright 1988 American Mathematical Society
- 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