Hostname: page-component-7d684dbfc8-kpkbf Total loading time: 0 Render date: 2023-09-25T23:17:55.277Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

The templates of non-singular Smale flows on three manifolds

Published online by Cambridge University Press:  24 May 2011

Department of Mathematics, Tongji University, Shanghai, 200092, China (email:


In this paper, we first discuss some connections between template theory and the description of basic sets of Smale flows on 3-manifolds due to F. Béguin and C. Bonatti. The main tools we use are symbolic dynamics, template moves and some combinatorial surgeries. Secondly, we obtain some relationship between the surgeries and the number of S1×S2 factors of M for a non-singular Smale flow on a given closed orientable 3-manifold M. We also prove that any template T can model a basic set Λ of a non-singular Smale flow on nS1×S2 for some positive integer n.

Research Article
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Asimov, D.. Round handles and non-singular Morse–Smale flows. Ann. of Math. (2) 102(1) (1975), 4154.CrossRefGoogle Scholar
[2]Béguin, F. and Bonatti, C.. Flots de Smale en dimension 3: presentations finies de voisinages invariants d’ensembles selles [Smale flows in dimension 3: finite presentations of invariant neighborhoods of saddle sets]. Topology 41(1) (2002), 119162 (in French).CrossRefGoogle Scholar
[3]Bowen, R.. One-dimensional hyperbolic sets for flows. J. Differential Equations 12 (1972), 173179.CrossRefGoogle Scholar
[4]Bowen, R. and Walters, P.. Expansive one-parameter flows. J. Differential Equations 12 (1972), 180193.CrossRefGoogle Scholar
[5]Birman, J. and Williams, R. F.. Knotted periodic orbits in dynamical systems. I. Lorenz’s equations. Topology 22(1) (1983), 4782.CrossRefGoogle Scholar
[6]Birman, J. and Williams, R. F.. Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots. Low-dimensional Topology (San Francisco, CA, 1981) (Contemporary Mathematics, 20). American Mathematical Society, Providence, RI, 1983, pp. 160.Google Scholar
[7]Franks, J.. Knots, links and symbolic dynamics. Ann. of Math. (2) 113(3) (1981), 529552.CrossRefGoogle Scholar
[8]Franks, J.. Homology and Dynamical Systems (CBMS, 49). American Mathematical Society, Providence, RI, 1982.CrossRefGoogle Scholar
[9]Franks, J.. Symbolic dynamics in flows on 3-manifolds. Trans. Amer. Math. Soc. 279(1) (1983), 231236.CrossRefGoogle Scholar
[10]Franks, J.. Flow equivalence of subshifts of finite type. Ergod. Th. & Dynam. Sys. 4(1) (1984), 5366.CrossRefGoogle Scholar
[11]Franks, J.. Nonsingular Smale flows on S 3. Topology 24(3) (1985), 265282.CrossRefGoogle Scholar
[12]Frank, G.. Templates and train tracks. Trans. Amer. Math. Soc. 308(2) (1988), 765784.CrossRefGoogle Scholar
[13]Ghrist, R. W., Holmes, P. J. and Sullivan, M. C.. Knots and Links in Three-dimensional Flows (Lecture Notes in Mathematics, 1654). Springer, Berlin, 1997.CrossRefGoogle Scholar
[14]Kauffman, L., Saito, M. and Sullivan, M.. Quantum invariants of templates. J. Knot Theory Ramifications 12(5) (2003), 653681.CrossRefGoogle Scholar
[15]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[16]Morgan, J.. Non-singular Morse–Smale flows on 3-dimensional manifolds. Topology 18 (1978), 4154.CrossRefGoogle Scholar
[17]Meleshuk, V.. Embedding templates in flows. PhD Thesis, Northwestern University, 2002.Google Scholar
[18]Parry, B. and Sullivan, D.. A topological invariant of flows on 1-dimensional spaces. Topology 14(4) (1975), 297299.CrossRefGoogle Scholar
[19]Pugh, C. and Shub, M.. Suspending subshifts. Contributions to Analysis and Geometry. Johns Hopkins University Press, Baltimore, MD, 1981, pp. 265275.Google Scholar
[20]de Rezende, K.. Smale flows on the three-sphere. Trans. Amer. Math. Soc. 303(1) (1987), 283310.CrossRefGoogle Scholar
[21]Robinson, C.. Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, 2nd edn.(Studies in Advanced Mathematics). CRC Press, Boca Raton, FL, 1999.Google Scholar
[22]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747817.CrossRefGoogle Scholar
[23]Sullivan, M. C.. Visually building Smale flows on S 3. Topology Appl. 106(1) (2000), 119.CrossRefGoogle Scholar
[24]Wada, M.. Closed orbits of nonsingular Morse–Smale flows on S 3. J. Math. Soc. Japan 41(3) (1989), 405413.CrossRefGoogle Scholar
[25]Williams, R.. Classification of subshifts of finite type. Ann. of Math. (2) 98 (1973), 120153; errata,Ann. of Math. (2) 99 (1974), 380–381.CrossRefGoogle Scholar
[26]Yu, B.. Lorenz like Smale flows on 3-manifolds. Topology Appl. 156(15) (2009), 24622469.CrossRefGoogle Scholar
[27]Yu, B.. Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds. Discrete Contin. Dyn. Syst. 29(3) (2011), 12771290; doi:10.3934/dcds.2011.29.1277.CrossRefGoogle Scholar