# American Institute of Mathematical Sciences

January  2016, 36(1): 509-540. doi: 10.3934/dcds.2016.36.509

## Behavior $0$ nonsingular Morse Smale flows on $S^3$

 1 Department of Mathematics, Tongji University, Shanghai 200092

Received  June 2014 Revised  December 2014 Published  June 2015

In this paper, we first develop the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as behavior $0$ NMS flows). It mainly contains the following several directions:
1. we use WLG to list behavior $0$ NMS flows on $S^3$;
2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows;
3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence;
4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.
Citation: Bin Yu. Behavior $0$ nonsingular Morse Smale flows on $S^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 509-540. doi: 10.3934/dcds.2016.36.509
##### References:
 [1] D. Asimov, Round handles and non-singular Morse-Smale flows, Ann. of Math. (2), 102 (1975), 41-54. doi: 10.2307/1970972. [2] D. Asimov, Homotopy of non-singular vector fields to structurally stable ones, Ann. of Math. (2), 102 (1975), 55-65. doi: 10.2307/1970973. [3] C. Bonatti, V. Grines and R. Langevin, Dynamical systems in dimension 2 and 3: conjugacy invariants and classification, in The geometry of differential equations and dynamical systems, Comput. Appl. Math., 20 (2001), 11-50. [4] B. Campos and P. Vindel, NMS flows on $S^3$ with no heteroclinic trajectories connecting saddle orbits, J. Dynam. Differential Equations, 24 (2012), 181-196. doi: 10.1007/s10884-012-9247-4. [5] J. Franks, Nonsingular Smale flows on $S^{3}$, Topology, 24 (1985), 265-282. doi: 10.1016/0040-9383(85)90002-3. [6] V. Grines and O. Pochinka, Morse-Smale cascades on 3-manifolds, (Russian) Uspekhi Mat. Nauk, 68 (2013), 129-188; translation in Russian Math. Surveys, 68 (2013), 117-173. [7] A. Hatcher, Notes on Basic 3-Manifold Topology,, Available from: , (). [8] J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1978), 41-53. doi: 10.1016/0040-9383(79)90013-2. [9] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982. [10] M. Peixoto, On the classification of flows on 2 -manifolds, in Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971, Academic Press, (1973), 389-419. [11] A. Prishlyak, A complete topological invariant of Morse-Smale flows and handle decompositions of 3-manifolds, (Russian) Fundam. Prikl. Mat., 11 (2005), 185-196; translation in J. Math. Sci. (N. Y.), 144 (2007), 4492-4499. doi: 10.1007/s10958-007-0287-y. [12] 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. [13] D. Rolfsen, Knots and Links, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990. [14] C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, $2^{nd}$ edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1999. [15] K. Sasano, Links of closed orbits of nonsingular Morse-Smale flows, Proc. Amer. Math. Soc., 88 (1983), 727-734. doi: 10.1090/S0002-9939-1983-0702309-0. [16] Y. Umanskii, Necessary and sufficient conditions for topological equivalence of three-dimensional dynamical Morse-Smale systems with a finite number of singular trajectories, (Russian)Mat. Sb., 181 (1990), 212-239; translation in Math. USSR-Sb., 69 (1991), 227-253. [17] M. Wada, Closed orbits of nonsingular Morse-Smale flows on $S^3$, J. Math. Soc. Japan, 41 (1989), 405-413. doi: 10.2969/jmsj/04130405. [18] K. Yano, The homotopy class of nonsingular Morse-Smale vector fields on 3 -manifolds, Invent. Math., 80 (1985), 435-451. doi: 10.1007/BF01388724. [19] B. Yu, The templates of non-singular Smale flows on three manifolds, Ergodic Theory Dynam. Systems, 32 (2012), 1137-1155. doi: 10.1017/S0143385711000150. [20] B. Yu, Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$, Trans. Amer. Math. Soc., 365 (2013), 767-783. doi: 10.1090/S0002-9947-2012-05636-4. [21] B. Yu, A note on homotopy classes of nonsingular verctor fields on $S^3$, C. R. Math. Acad. Sci. Paris, 352 (2014), 351-355. doi: 10.1016/j.crma.2014.01.016.

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##### References:
 [1] D. Asimov, Round handles and non-singular Morse-Smale flows, Ann. of Math. (2), 102 (1975), 41-54. doi: 10.2307/1970972. [2] D. Asimov, Homotopy of non-singular vector fields to structurally stable ones, Ann. of Math. (2), 102 (1975), 55-65. doi: 10.2307/1970973. [3] C. Bonatti, V. Grines and R. Langevin, Dynamical systems in dimension 2 and 3: conjugacy invariants and classification, in The geometry of differential equations and dynamical systems, Comput. Appl. Math., 20 (2001), 11-50. [4] B. Campos and P. Vindel, NMS flows on $S^3$ with no heteroclinic trajectories connecting saddle orbits, J. Dynam. Differential Equations, 24 (2012), 181-196. doi: 10.1007/s10884-012-9247-4. [5] J. Franks, Nonsingular Smale flows on $S^{3}$, Topology, 24 (1985), 265-282. doi: 10.1016/0040-9383(85)90002-3. [6] V. Grines and O. Pochinka, Morse-Smale cascades on 3-manifolds, (Russian) Uspekhi Mat. Nauk, 68 (2013), 129-188; translation in Russian Math. Surveys, 68 (2013), 117-173. [7] A. Hatcher, Notes on Basic 3-Manifold Topology,, Available from: , (). [8] J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1978), 41-53. doi: 10.1016/0040-9383(79)90013-2. [9] J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982. [10] M. Peixoto, On the classification of flows on 2 -manifolds, in Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971, Academic Press, (1973), 389-419. [11] A. Prishlyak, A complete topological invariant of Morse-Smale flows and handle decompositions of 3-manifolds, (Russian) Fundam. Prikl. Mat., 11 (2005), 185-196; translation in J. Math. Sci. (N. Y.), 144 (2007), 4492-4499. doi: 10.1007/s10958-007-0287-y. [12] 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. [13] D. Rolfsen, Knots and Links, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990. [14] C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, $2^{nd}$ edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1999. [15] K. Sasano, Links of closed orbits of nonsingular Morse-Smale flows, Proc. Amer. Math. Soc., 88 (1983), 727-734. doi: 10.1090/S0002-9939-1983-0702309-0. [16] Y. Umanskii, Necessary and sufficient conditions for topological equivalence of three-dimensional dynamical Morse-Smale systems with a finite number of singular trajectories, (Russian)Mat. Sb., 181 (1990), 212-239; translation in Math. USSR-Sb., 69 (1991), 227-253. [17] M. Wada, Closed orbits of nonsingular Morse-Smale flows on $S^3$, J. Math. Soc. Japan, 41 (1989), 405-413. doi: 10.2969/jmsj/04130405. [18] K. Yano, The homotopy class of nonsingular Morse-Smale vector fields on 3 -manifolds, Invent. Math., 80 (1985), 435-451. doi: 10.1007/BF01388724. [19] B. Yu, The templates of non-singular Smale flows on three manifolds, Ergodic Theory Dynam. Systems, 32 (2012), 1137-1155. doi: 10.1017/S0143385711000150. [20] B. Yu, Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$, Trans. Amer. Math. Soc., 365 (2013), 767-783. doi: 10.1090/S0002-9947-2012-05636-4. [21] B. Yu, A note on homotopy classes of nonsingular verctor fields on $S^3$, C. R. Math. Acad. Sci. Paris, 352 (2014), 351-355. doi: 10.1016/j.crma.2014.01.016.
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