Graphs of NMS flows on \(S^{3}\) with knotted saddle orbits and no heteroclinic trajectories. (English) Zbl 1137.37312

Campos, B.; Vindel, P.

Acta Math. Sin., Engl. Ser. 23, No. 12, 2213-2224 (2007).

Summary: We consider NMS systems on \(S^{3}\) without heteroclinic trajectories connecting two saddles orbits and we build the dual graphs associated with this kind of flows. We prove that flows coming from essential attachments of orientable round 1-handles can be reproduced from the corresponding dual graph.

MSC:

37D15	Morse-Smale systems
37A20	Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
57M25	Knots and links in the \(3\)-sphere (MSC2010)
37E25	Dynamical systems involving maps of trees and graphs

Keywords:

Morse-Smale flows; Wada operations; invariants topological equivalence classes; dual graph

References:

[1]	Peixoto, M. M.: On the Classification of Flows on 2-Manifolds, Proc. Symp. Dyn. Syst. Brazil 1971. Academic Press, 1973, 389–419
[2]	Wang, X. L.: The C*-algebras of Morse-Smale flows on two-manifolds. Ergod. Th. and Dynam. Sys., 10, 565–597 (1989) · Zbl 0728.58017
[3]	Franks, J.: Non Singular Smale Flows on S 3. Topology, 24(3), 265–282 (1985) · Zbl 0609.58039
[4]	Bolsinov, A. V., Matveev, S. V., Fomenko, A. T.: Topological classification of integrable Hamiltonian systems with two degrees of freedom. A list of systems of small complexity. Russian Math, Surveys, 45(2), 59–94 (1990) · Zbl 0705.58025
[5]	Nguen T’en, Z., Fomenko, A. T.: Topological classification of integrable non-degenerate Hamiltonians on a constant energy three-dimensional sphere. Russian Math, Surveys, 45(6), 109–135 (1990) · Zbl 0721.58022
[6]	Asimov, D.: Round handles and Non-singular Morse-Smale flows. Ann. Math., 102, 41–54 (1975) · Zbl 0316.57020
[7]	Morgan, J.: Non Singular Morse–Smale flows on 3-dimensional manifolds. Topology, 18, 41–53 (1978) · Zbl 0406.58020
[8]	Wada, M.: Closed orbits of non-singular Morse-Smale flows on S 3. J. Math. Soc. Japan, 41(3), 405–413 (1989) · Zbl 0672.58042
[9]	Campos, B., Martínez Alfaro, J., Vindel, P.: Graphs of NMS Flows on S 3 with unknotted saddle periodic orbits. Topology Proc, 26(2), 469–483 (2003) · Zbl 1082.37503
[10]	Chartrand, G., Lesniak, L.: Graphs and digraphs. Ed. Chapman & Hall, 1996 · Zbl 0890.05001