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Lorenz knots are prime

Published online by Cambridge University Press:  19 September 2008

R. F. Williams
R. F. Williams
Affiliation:
Mathematics Department, Northwestern University, Evanston, IL 60201, USA
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Abstract

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Lorenz knots are the periodic orbits of a certain geometrically defined differential equation in ℝ3. This is called the ‘geometric Lorenz attractor’ as it is only conjecturally the real Lorenz attractor. These knots have been studied by the author and Joan Birman via a ‘knot-holder’, i.e. a certain branched two-manifold H. To show such knots are prime we suppose the contrary which implies the existence of a splitting sphere, S2. The technique of the proof is to study the intersection S2∩H. A novelty here is that S2∩H is likewise branched.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 1 , March 1984 , pp. 147 - 163
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Birman, J. & Williams, R.. Knotted periodic orbits I: Lorenz knots. Topology 22 (1983), 47–82.CrossRefGoogle Scholar
[2]Birman, J. & Williams, R.. Knotted periodic orbits II: Fibered knots. Low dimensional Topology, vol. 20, Contemporary Mathematics. Amer. Math. Soc: Providence, R.I. 1983.CrossRefGoogle Scholar
[3]Guckenheimer, J.. A strange, strange attractor. The Hopf Bifurcation (ed. Marsden, and McCracken, ). Appl. Math. Sci., Springer-Verlag, 1976.Google Scholar
[4]Guckenheimer, J. & Williams, R.. Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72.CrossRefGoogle Scholar
[5]Williams, R. F.. The structure of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math. 50 (1979) 73–100.CrossRefGoogle Scholar