# American Institute of Mathematical Sciences

July  2011, 29(3): 1277-1290. doi: 10.3934/dcds.2011.29.1277

## Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds

 1 Department of Mathematics, Tongji University, Shanghai 200092, China

Received  March 2010 Revised  June 2010 Published  November 2010

In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF which has a regular level set homeomorphic to $(n+1)T^{2}$ $(n\in \mathbb{Z}, n\geq 0)$ if and only if $M=M'$#$n S^{1}\times S^{2}$. Then we discuss how to realize a template as a basic set of an NSF on a 3-manifold. We focus on the connection between the genus of the template $T$ and the topological structure of the realizing 3-manifold $M$.
Citation: Bin Yu. Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1277-1290. doi: 10.3934/dcds.2011.29.1277
##### References:
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##### References:
 [1] F. Béguin and C. Bonatti, Flots de Smale en dimension 3: Présentations finies de voisinages invariants d'ensembles selles, (French) [Smale flows in dimension 3: Finite presentations of invariant neighborhoods of saddle sets], Topology, 41 (2002), 119-162. doi: 10.1016/S0040-9383(00)00032-X.  Google Scholar [2] R. Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations, 12 (1972), 173-179. doi: 10.1016/0022-0396(72)90012-5.  Google Scholar [3] J. Birman and R. F. Williams, Knotted periodic orbits in dynamical systems. I. Lorenz's equations, Topology, 22 (1983), 47-82. doi: 10.1016/0040-9383(83)90045-9.  Google Scholar [4] J. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, in "Low-Dimensional Topology" (San Francisco, Calif., 1981), Contemp. Math., 20, Amer. Math. Soc., Providence, RI, (1983), 1-60.  Google Scholar [5] R. N. Cruz and K. A. de Rezende, Cycle rank of Lyapunov graphs and the genera of manifolds, Proc. Amer. Math. Soc, 126 (1998), 3715-3720. doi: 10.1090/S0002-9939-98-04957-0.  Google Scholar [6] J. Franks, Nonsingular Smale flows on $S^{3}$, Topology, 24 (1985), 265-282. doi: 10.1016/0040-9383(85)90002-3.  Google Scholar [7] J. Franks, Symbolic dynamics in flows on three-manifolds, Trans. Amer. Math. Soc, 279 (1983), 231-236. doi: 10.1090/S0002-9947-1983-0704612-1.  Google Scholar [8] J. Franks, "Homology and Dynamical Systems," CBMS 49, American Mathematical Society, Providence, Rhode Island, 1982.  Google Scholar [9] J. Franks, Knots, links and symbolic dynamics, Ann. of Math. (2), 113 (1981), 529-552. doi: 10.2307/2006996.  Google Scholar [10] S. R. Fenley, Anosov flows in 3-manifolds, Ann. of Math. (2), 139 (1994), 79-115. doi: 10.2307/2946628.  Google Scholar [11] G. Frank, Templates and train tracks, Trans. Amer. Math. Soc, 308 (1988), 765-784. doi: 10.1090/S0002-9947-1988-0951627-9.  Google Scholar [12] R. W. Ghrist, P. J. Holmes and M. C. Sullivan, "Knots and Links in Three-dimensional Flows," Lecture Notes in Mathematics, 1654, Springer-Verlag, Berlin, 1997.  Google Scholar [13] J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1978), 41-54. doi: 10.1016/0040-9383(79)90013-2.  Google Scholar [14] V. Meleshuk, "Embedding Templates in Flows," Ph.D thesis, Northwestern University, 2002. Google Scholar [15] N. Oka, Notes on Lyapunov graphs and nonsingular Smale flows on three manifolds, Nagoya Math. J, 117 (1990), 37-61.  Google Scholar [16] C. Pugh and M. Shub, Suspending subshifts, in "Contributions to Analysis and Geometry" (eds. Md. Baltimore), Johns Hopkins Univ. Press, (1981), 265-275.  Google Scholar [17] K. de Rezende, Smale flows on the three-sphere, Trans. Amer. Math. Soc, 303 (1987), 283-310. doi: 10.1090/S0002-9947-1987-0896023-7.  Google Scholar [18] C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos," 2nd edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999.  Google Scholar [19] D. Rolfsen, "Knots and Links," Publish or Perish, Inc., Berkeley, CA, 1976.  Google Scholar [20] M. Saito, On closed orbits of Morse-Smale flows on 3-manifolds, Bull. London Math. Soc, 23 (1991), 482-486. doi: 10.1112/blms/23.5.482.  Google Scholar [21] M. C. Sullivan, Visually building Smale flows on $S^{3}$, Topology Appl, 106 (2000), 1-19. doi: 10.1016/S0166-8641(99)00069-3.  Google Scholar [22] B. Yu, Lorenz like Smale flows on three-manifolds, Topology Appl, 156 (2009), 2462-2469. doi: 10.1016/j.topol.2009.07.008.  Google Scholar
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