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 - A, 2011, 29 (3) : 1277-1290. doi: 10.3934/dcds.2011.29.1277
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. 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. 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. 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, 20 (1983), 1. 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. doi: 10.1090/S0002-9939-98-04957-0. Google Scholar

[6]

J. Franks, Nonsingular Smale flows on $S^{3}$,, Topology, 24 (1985), 265. 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. doi: 10.1090/S0002-9947-1983-0704612-1. Google Scholar

[8]

J. Franks, "Homology and Dynamical Systems,", CBMS \textbf{49}, 49 (1982). Google Scholar

[9]

J. Franks, Knots, links and symbolic dynamics,, Ann. of Math. (2), 113 (1981), 529. doi: 10.2307/2006996. Google Scholar

[10]

S. R. Fenley, Anosov flows in 3-manifolds,, Ann. of Math. (2), 139 (1994), 79. doi: 10.2307/2946628. Google Scholar

[11]

G. Frank, Templates and train tracks,, Trans. Amer. Math. Soc, 308 (1988), 765. 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 (1997). Google Scholar

[13]

J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds,, Topology, 18 (1978), 41. doi: 10.1016/0040-9383(79)90013-2. Google Scholar

[14]

V. Meleshuk, "Embedding Templates in Flows,", Ph.D thesis, (2002). Google Scholar

[15]

N. Oka, Notes on Lyapunov graphs and nonsingular Smale flows on three manifolds,, Nagoya Math. J, 117 (1990), 37. Google Scholar

[16]

C. Pugh and M. Shub, Suspending subshifts,, in, (1981), 265. Google Scholar

[17]

K. de Rezende, Smale flows on the three-sphere,, Trans. Amer. Math. Soc, 303 (1987), 283. doi: 10.1090/S0002-9947-1987-0896023-7. Google Scholar

[18]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,", 2nd edition, (1999). Google Scholar

[19]

D. Rolfsen, "Knots and Links,", Publish or Perish, (1976). Google Scholar

[20]

M. Saito, On closed orbits of Morse-Smale flows on 3-manifolds,, Bull. London Math. Soc, 23 (1991), 482. 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. 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. doi: 10.1016/j.topol.2009.07.008. Google Scholar

show all references

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. 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. 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. 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, 20 (1983), 1. 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. doi: 10.1090/S0002-9939-98-04957-0. Google Scholar

[6]

J. Franks, Nonsingular Smale flows on $S^{3}$,, Topology, 24 (1985), 265. 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. doi: 10.1090/S0002-9947-1983-0704612-1. Google Scholar

[8]

J. Franks, "Homology and Dynamical Systems,", CBMS \textbf{49}, 49 (1982). Google Scholar

[9]

J. Franks, Knots, links and symbolic dynamics,, Ann. of Math. (2), 113 (1981), 529. doi: 10.2307/2006996. Google Scholar

[10]

S. R. Fenley, Anosov flows in 3-manifolds,, Ann. of Math. (2), 139 (1994), 79. doi: 10.2307/2946628. Google Scholar

[11]

G. Frank, Templates and train tracks,, Trans. Amer. Math. Soc, 308 (1988), 765. 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 (1997). Google Scholar

[13]

J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds,, Topology, 18 (1978), 41. doi: 10.1016/0040-9383(79)90013-2. Google Scholar

[14]

V. Meleshuk, "Embedding Templates in Flows,", Ph.D thesis, (2002). Google Scholar

[15]

N. Oka, Notes on Lyapunov graphs and nonsingular Smale flows on three manifolds,, Nagoya Math. J, 117 (1990), 37. Google Scholar

[16]

C. Pugh and M. Shub, Suspending subshifts,, in, (1981), 265. Google Scholar

[17]

K. de Rezende, Smale flows on the three-sphere,, Trans. Amer. Math. Soc, 303 (1987), 283. doi: 10.1090/S0002-9947-1987-0896023-7. Google Scholar

[18]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,", 2nd edition, (1999). Google Scholar

[19]

D. Rolfsen, "Knots and Links,", Publish or Perish, (1976). Google Scholar

[20]

M. Saito, On closed orbits of Morse-Smale flows on 3-manifolds,, Bull. London Math. Soc, 23 (1991), 482. 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. 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. doi: 10.1016/j.topol.2009.07.008. Google Scholar

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