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Vector fields with transverse foliations, II

Published online by Cambridge University Press:  19 September 2008

Sue Goodman
Sue Goodman
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514, USA
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Abstract

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When does a non-singular flow on a 3-manifold have a 2-dimensional foliation everywhere transverse to it? A complete answer is given for a large class of flows, those with 1-dimensional hyperbolic chain recurrent set. We find a simple necessary and sufficient condition on the linking of periodic orbits of the flow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 2 , June 1986 , pp. 193 - 203
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[A-G]Asimov, D. & Goodman, S.. Stable non-singular flows not transverse to any foliation. Preprint, 1978.Google Scholar
[B-W]Birman, J. & Williams, R. F.. Knotted periodic orbits in dynamical systems II: Knot holders for fibred knots. In Low Dimensional Topology. Amer. Math. Soc. (1983), 1–60.Google Scholar
[B]Bowen, R.. One-dimensional hyperbolic sets for flows. J. Diff. Eq. 12 (1972), 173–79.CrossRefGoogle Scholar
[E-H-N]Eisenbud, D., Hirsch, U. & Neumann, W.. Transverse foliations of Seifert bundles and self-homeomorphisms of the circle. Comm. Math. Helv. 56 (1981), 638–660.CrossRefGoogle Scholar
[Fra1]Franks, J.. Homology and Dynamical Systems. CBMS Regional Conf. Series in Math. 49, Amer. Math. Soc.Google Scholar
[Fra2]Franks, J.. Non-singular Smale flow on S3. Topology 24 (1985), 265–282.CrossRefGoogle Scholar
[Fri]Fried, D.. Cross-sections to flows. Topology 21 (1982), 353–372.CrossRefGoogle Scholar
[G]Goodman, S.. Vector fields with transverse foliations. Topology 24 (1985), 333–340.CrossRefGoogle Scholar
[M]Morgan, J.. Non-singular Morse-Smale flows on 3-dimensional manifolds. Topology 18 (1978), 41–53.CrossRefGoogle Scholar
[Ne]Newhouse, S.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. I.H.E.S. 50 (1979), 101–152.Google Scholar
[No]Novikov, S. P.. Topology of foliations. Trans. Moscow Math. Soc. 14 (1965), 268–304.Google Scholar
[O]Oliveira, M.. C0-density of structurally stable dynamical systems. Thesis, Univ. of Warwick, 1976 (revised 1979).Google Scholar
[R]Reinhart, B.. Cobordism and the Euler number. Topology 2 (1963), 173–177.CrossRefGoogle Scholar
[Sch]Schwartzman, S.. Asymptotic cycles. Annals of Math. 66 (1957), 270–283.CrossRefGoogle Scholar
[Sm]Smale, S.. Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[Th]Thurston, W.. A local construction of foliations for 3-manifolds. In Amer. Math. Soc. Proc. Symp. Pure Math. 27 (1975), 315–319.CrossRefGoogle Scholar
[W]Wood, J.. Bundles with totally disconnected structure group. Comm. Math. Helv. 46 (1971), 257–279.CrossRefGoogle Scholar
[Z]Zeeman, E. C.. Morse inequalities for difleomorphisms with shoes and flows with solenoids. In Dynamical Systems - Warwick 1974. Lecture Notes in Maths. 468, Springer (1975), 44–48.Google Scholar