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

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

Bin Yu 1,

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.
Keywords: topologically equivalent classes., links, Behavior $0$ nonsingular Morse-Smale flows, weighted Lyapunov graphs.
Mathematics Subject Classification: Primary: 37D15, 37C15; Secondary: 37C15, 57M2.
Citation: Bin Yu. Behavior $0$ nonsingular Morse Smale flows on $S^3$. Discrete & Continuous Dynamical Systems - A, 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. doi: 10.2307/1970972. Google Scholar

[2]

D. Asimov, Homotopy of non-singular vector fields to structurally stable ones,, Ann. of Math. (2), 102 (1975), 55. doi: 10.2307/1970973. Google Scholar

[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, 20 (2001), 11. Google Scholar

[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. doi: 10.1007/s10884-012-9247-4. Google Scholar

[5]

J. Franks, Nonsingular Smale flows on $S^{3}$,, Topology, 24 (1985), 265. doi: 10.1016/0040-9383(85)90002-3. Google Scholar

[6]

V. Grines and O. Pochinka, Morse-Smale cascades on 3-manifolds,, (Russian) Uspekhi Mat. Nauk, 68 (2013), 129. Google Scholar

[7]

A. Hatcher, Notes on Basic 3-Manifold Topology,, Available from: , (). Google Scholar

[8]

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

[9]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction,, Translated from the Portuguese by A. K. Manning, (1982). Google Scholar

[10]

M. Peixoto, On the classification of flows on 2 -manifolds,, in Dynamical systems (Proc. Sympos., (1973), 389. Google Scholar

[11]

A. Prishlyak, A complete topological invariant of Morse-Smale flows and handle decompositions of 3-manifolds,, (Russian) Fundam. Prikl. Mat., 11 (2005), 185. doi: 10.1007/s10958-007-0287-y. Google Scholar

[12]

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

[13]

D. Rolfsen, Knots and Links,, Corrected reprint of the 1976 original. Mathematics Lecture Series, (1976). Google Scholar

[14]

C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,, $2^{nd}$ edition, (1999). Google Scholar

[15]

K. Sasano, Links of closed orbits of nonsingular Morse-Smale flows,, Proc. Amer. Math. Soc., 88 (1983), 727. doi: 10.1090/S0002-9939-1983-0702309-0. Google Scholar

[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. Google Scholar

[17]

M. Wada, Closed orbits of nonsingular Morse-Smale flows on $S^3$,, J. Math. Soc. Japan, 41 (1989), 405. doi: 10.2969/jmsj/04130405. Google Scholar

[18]

K. Yano, The homotopy class of nonsingular Morse-Smale vector fields on 3 -manifolds,, Invent. Math., 80 (1985), 435. doi: 10.1007/BF01388724. Google Scholar

[19]

B. Yu, The templates of non-singular Smale flows on three manifolds,, Ergodic Theory Dynam. Systems, 32 (2012), 1137. doi: 10.1017/S0143385711000150. Google Scholar

[20]

B. Yu, Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$,, Trans. Amer. Math. Soc., 365 (2013), 767. doi: 10.1090/S0002-9947-2012-05636-4. Google Scholar

[21]

B. Yu, A note on homotopy classes of nonsingular verctor fields on $S^3$,, C. R. Math. Acad. Sci. Paris, 352 (2014), 351. doi: 10.1016/j.crma.2014.01.016. Google Scholar

show all references

References:
[1]

D. Asimov, Round handles and non-singular Morse-Smale flows,, Ann. of Math. (2), 102 (1975), 41. doi: 10.2307/1970972. Google Scholar

[2]

D. Asimov, Homotopy of non-singular vector fields to structurally stable ones,, Ann. of Math. (2), 102 (1975), 55. doi: 10.2307/1970973. Google Scholar

[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, 20 (2001), 11. Google Scholar

[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. doi: 10.1007/s10884-012-9247-4. Google Scholar

[5]

J. Franks, Nonsingular Smale flows on $S^{3}$,, Topology, 24 (1985), 265. doi: 10.1016/0040-9383(85)90002-3. Google Scholar

[6]

V. Grines and O. Pochinka, Morse-Smale cascades on 3-manifolds,, (Russian) Uspekhi Mat. Nauk, 68 (2013), 129. Google Scholar

[7]

A. Hatcher, Notes on Basic 3-Manifold Topology,, Available from: , (). Google Scholar

[8]

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

[9]

J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction,, Translated from the Portuguese by A. K. Manning, (1982). Google Scholar

[10]

M. Peixoto, On the classification of flows on 2 -manifolds,, in Dynamical systems (Proc. Sympos., (1973), 389. Google Scholar

[11]

A. Prishlyak, A complete topological invariant of Morse-Smale flows and handle decompositions of 3-manifolds,, (Russian) Fundam. Prikl. Mat., 11 (2005), 185. doi: 10.1007/s10958-007-0287-y. Google Scholar

[12]

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

[13]

D. Rolfsen, Knots and Links,, Corrected reprint of the 1976 original. Mathematics Lecture Series, (1976). Google Scholar

[14]

C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,, $2^{nd}$ edition, (1999). Google Scholar

[15]

K. Sasano, Links of closed orbits of nonsingular Morse-Smale flows,, Proc. Amer. Math. Soc., 88 (1983), 727. doi: 10.1090/S0002-9939-1983-0702309-0. Google Scholar

[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. Google Scholar

[17]

M. Wada, Closed orbits of nonsingular Morse-Smale flows on $S^3$,, J. Math. Soc. Japan, 41 (1989), 405. doi: 10.2969/jmsj/04130405. Google Scholar

[18]

K. Yano, The homotopy class of nonsingular Morse-Smale vector fields on 3 -manifolds,, Invent. Math., 80 (1985), 435. doi: 10.1007/BF01388724. Google Scholar

[19]

B. Yu, The templates of non-singular Smale flows on three manifolds,, Ergodic Theory Dynam. Systems, 32 (2012), 1137. doi: 10.1017/S0143385711000150. Google Scholar

[20]

B. Yu, Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$,, Trans. Amer. Math. Soc., 365 (2013), 767. doi: 10.1090/S0002-9947-2012-05636-4. Google Scholar

[21]

B. Yu, A note on homotopy classes of nonsingular verctor fields on $S^3$,, C. R. Math. Acad. Sci. Paris, 352 (2014), 351. doi: 10.1016/j.crma.2014.01.016. Google Scholar

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