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

Bin Yu

doi:10.3934/dcds.2016.36.509

Article Contents

2016, Volume 36, Issue 1: 509-540. Doi: 10.3934/dcds.2016.36.509

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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: May 2017

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Abstract

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:

Behavior $0$ nonsingular Morse-Smale flows,
weighted Lyapunov graphs,
links,
topologically equivalent classes.

Mathematics Subject Classification: Primary: 37D15, 37C15; Secondary: 37C15, 57M25.

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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: http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html
[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.