Behavior 0 nonsingular Morse Smale flows on S^3

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

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

1.	Department of Mathematics, Tongji University, Shanghai 200092

Received June 2014 Revised December 2014 Published June 2015

Abstract
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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*

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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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